This checks the time required to compute invariants[s], for s = 2, 3, ..., 8  
(about 13 minutes with a 2.4 MegaHerz PC).
The corresponding relative invariants are expressed
as differential-polynomial combinations of the basic ones.

The version number of Mathematica used to evaluate this notebook:

$VersionNumber

The main results are summarized at the end of this notebook.

u[s_, p_, q_, r_] := 
        Floor[ (1/(p + q + r))*
            (s - Sum[(p + q + k)*e[s, p, q, k], {k, r + 1, s - p - q}]
                   - Sum[(p + j + k)*e[s, p, j, k], {j, q + 1, 2}, {k, 0, 
                    s - p - j}] - 
                Sum[(i + j + k)*e[s, i, j, k], {i, p + 1, 2}, {j, i, 2}, {
                    k, 0, s - i - j}])]

nM[s_] := 
        Module[{counter, seq}, counter = 0; 
          seq =  Apply[Sequence, 
              Reverse[Flatten[
                  Table[{e[s, i, j, k], 0, u[s, i, j, k]}, {i, 0, 2}, {j, 
                      Max[1, i], 2}, {k, 0, s - i - j}], 2]]]; 
          Do[If[(
                  Sum[(i + j + k)*e[s, i, j, k], {i, 0, 2}, {j, 
                      Max[1, i], 2}, {k, 0, s - i - j}]) == s, 
              (counter = counter + 1; 
                monM[s, counter] = 
                  Product[
                    D[w[i, j][z], {z, k}]^e[s, i, j, k], {i, 0, 2}, {j, 
                      Max[1, i], 2}, {k, 0, s - i - j}] ), 
              counter = counter], Evaluate[seq]]; counter]

printM[s_] := 
        Do[ Print["monM[", s, ", ", i, "] = ", monM[s, i]], {i, 
            1, nM[s]}]

wPolynomial[s_]:=Sum[x[s, i]*monM[s, i],{i,1,nM[s]}]

sub[s_]:=Flatten[ Table[D[w[i, j][z], {z, k}] -> D[c[i, j][z], {z, k}],{i, 0, 2},{j, Max[1, i], 2},{k,0,s-i-j}] ]

cPoly[s_] := wPolynomial[s] /. sub[s]

m = 2;  c[0, 0][z_] = 1; 
      c[j_, i_][z_] := c[i, j][z] /; j > i;

cOneStar[i_, j_][z_] := 
        Sum[Binomial[m - mu, i - mu]*Binomial[m - nu, j - nu]*
            D[rho[z], {z, i - mu}]*D[rho[z], {z, j - nu}]/rho[z]^2*
            c[mu, nu][z], {mu, 0, i}, {nu, 0, j}];

sub1[s_] :=  
        Flatten[ 
          Table[D[w[i, j][z], {z, k}] -> 
              D[cOneStar[i, j][z], {z, k}], {i, 0, 2}, {j, Max[1, i], 
              2}, {k, 0, s - i - j} ] ]

\(cOneStarPoly[s_] := wPolynomial[s] /. sub1[s]
\)

alpha[0][j_, zet_] = 1; 
      g[zet_] = f''[zet]/f'[zet];

alpha[1][j_, zet_]= g[zet]*Sum[-k, {k, 1, j}];

alpha[2][j_, zet_]=(D[g[zet], zet]*Sum[-k*(k + 1)/2, {k, 1, j}]+g[zet]^2*Sum[(1 + k)^2*k/2, {k, 1, j}] )//Simplify;

cTwoStar[i_, j_][zet_] := 
        Sum[f'[zet]^(mu + nu)*alpha[i - mu][m - i, zet]*
            alpha[j - nu][m - j, zet]*c[mu, nu][f[zet]], {mu, 0, i}, {
            nu, 0, j}]

sub2[s_] := 
        Flatten[ 
          Table[D[w[i, j][z], {z, k}] -> 
              D[cTwoStar[i, j][zet], {zet, k}], {i, 0, 2}, {j, Max[1, i], 
              2}, {k, 0, s - i - j} ] ]

cTwoStarPoly[s_] := wPolynomial[s] /. sub2[s]

firstCondition[s_]:=firstCondition[s] = Expand[rho[z]^s*(cOneStarPoly[s] - cPoly[s])];

secondCondition[s_]:=secondCondition[s] = Expand[f'[zet]^s*(cTwoStarPoly[s] - 
                    f'[zet]^s*(cPoly[s] /. z -> f[zet]))];

v[s_, p_] := 
        Floor[(1/p)*
            (s - Sum[
                  (i + j + k)*e[s, i, j, k], {i, 0, 2}, {j, Max[1, i], 
                    2}, {k, 0, s - i - j}] - Sum[t*h[s, t], {t, p + 1, s}])]

nA[s_]:=Module[{counter, seq1, seq2},counter = 0;seq1 = 
                Apply[Sequence, 
                  Reverse[
                    Flatten[
                      Table[{e[s, i, j, k], 0, u[s, i, j, k]}, {i, 0, 2}, {j, 
                          Max[1, i], 2}, {k, 0, s - i - j}], 2]]];seq2 = 
                Apply[Sequence, 
                  Reverse[Table[{h[s, t], 0, v[s, t]}, {t, 1, s}]]];Do[If[(Sum[
                              (i + j + k)*e[s, i, j, k], {i, 0, 2}, {j, 
                                Max[1, i], 2}, {k, 0, s - i - j}] + 
                            Sum[t*h[s, t], {t, 1, s}]) == s,(counter = counter + 1;monA[s, counter]=Product[
                                D[c[i, j][z], {z, k}]^e[s, i, j, k], {i, 
                                  0, 2}, {j, Max[1, i], 2}, {k, 0, s - i - j}]
                                *rho[z]^(s - Sum[h[s, t], {t, 1, s}])*Product[D[rho[z], {z, t}]^h[s, t],{t, 1, s}]),counter = counter],Evaluate[seq1],Evaluate[seq2]];counter]

printA[s_] := 
        Do[Print["monA[", s, ", ", i, "] = ", monA[s, i]], {i, 1, 
            nA[s]}]

printA[2]

nB[s_]:=Module[{counter, seq1, seq2},counter = 0;seq1 = 
                Apply[Sequence, 
                  Reverse[
                    Flatten[
                      Table[{e[s, i, j, k], 0, u[s, i, j, k]}, {i, 0, 2}, {j, 
                          Max[1, i], 2}, {k, 0, s - i - j}], 2]]];seq2 = 
                Apply[Sequence, 
                  Reverse[Table[{h[s, t], 0, v[s, t]}, {t, 1, s}]]];Do[If[(Sum[
                              (i + j + k)*e[s, i, j, k], {i, 0, 2}, {j, 
                                Max[1, i], 2}, {k, 0, s - i - j}] +  
                            Sum[t*h[s, t], {t, 1, s}]) == s,(counter = counter + 1;monB[s, counter]=Product[
                                (D[c[i, j][z], {z, k}] /. z -> f[zet] )
                                    ^e[s, i, j, k], {i, 0, 2}, {j, 
                                  Max[1, i], 2}, {k, 0, s - i - j}]*f'[zet]^(2 s - Sum[(t + 1)*h[s, t], {t, 1, s}])
                                  *Product[D[f[zet], {zet, t + 1}]^h[s, t],{t, 1, s}]),counter = counter],Evaluate[seq1],Evaluate[seq2]];counter]

printB[s_] := 
        Do[ Print["monB[", s, ", ", i, "] = ", monB[s, i]], {i, 
            1, nB[s]}]

firstCoefficients[s_] := 
        Do[a[s, i] = Coefficient[firstCondition[s], monA[s, i]], {i, 1, 
            nA[s]}]

firstCoefficients[4]

firstZeroCheck[s_] := 
        Expand[firstCondition[s] - Sum[a[s, i]*monA[s, i], {i, 1, nA[s]}]]

secondCoefficients[s_] := 
        Do[ b[s, j] = Coefficient[secondCondition[s], monB[s, j]], {j, 1, 
            nB[s]}]

secondCoefficients[4]

secondZeroCheck[s_] := 
        Expand[secondCondition[s] - Sum[ b[s, j]*monB[s, j], {j, 1, nB[s]}]]

eqs[s_] := 
        Join[Table[a[s, i] == 0, {i, 1, nA[s]}], 
          Table[b[s, j] == 0, {j, 1, nB[s]}]]

rules[s_] := Flatten[Solve[eqs[s], Table[x[s, i], {i, 1, nM[s]}]]]

(*  Print["firstZeroCheck[", s, "] = ", firstZeroCheck[s]];   *)

(* Print["secondZeroCheck[", s, "] = ", secondZeroCheck[s]]; 
         *)

invariants[s_]:=Module[{expr, counter},counter = 0;
		Print["   "];
		Print["   Computation of invariants[", s, "]"]
              ;
		nM[s];Print["Number of Monic Monomials = ", nM[s]];
	nA[s];Print["Number of firstCondition Terms = ", nA[s]];
  nB[s]; Print["Number of secondCondition Terms = ", nB[s]];
		firstCondition[s];  firstCoefficients[s];
		Print["firstZeroCheck[", s, "] = ",  
              firstZeroCheck[s]];
		secondCondition[s]; secondCoefficients[s];
			Print["secondZeroCheck[", s, "] = ", 
              secondZeroCheck[s] ]; 
		eqs[s];    rules[s]; 
		expr = 
                Expand[(Sum[ x[s, i]*monM[s, i], {i, 1, nM[s]}]) /. 
                    rules[s]];Do[ If[Length[Coefficient[expr, x[s, i]]] != 0, 
                  (counter = counter + 1; 
                    Inv[s, counter] = Coefficient[expr, x[s, i]]; 
                    Print["Inv[", s, ", ", counter, "] = ", 
                      Inv[s, counter]] ),  counter = counter], {i, 1, 
                  nM[s]}]; n[s] = counter; Print["n[", s, "] = ",  n[s] ]   ]

Date[]

Do[ (  invariants[i];  Print[ Date[] ] ),  {i, 2, 
        8} ]

Date[]

m  =  2

\(\(\(x[2, 2]\)[
          z_]\  = \ \((\ \ \(Invar[m, 2, 2]\)[
              z]\  + \ \((2 \((m - 1)\)/m)\)*\(w[0, 1]\)[z]*\(w[1, 2]\)[
                z]
			 - \ \((6 \((m - 1)\)/\((5 \((m + 1)\) m)\))\) \(w[
                  0, 2]\)[z]*\(w[1, 1]\)[z]
			 + \ \(w[0, 2]\)[
                z]^2
		 - \((\((5  m + 
                      2)\) \((m - 1)\)^2/\((5 \((m + 1)\) m^2)\))\) \((\(w[0, 
                      1]\)[z])\)^2*\(w[1, 1]\)[
                z]\ 
		 - \ \((2 \((5  m + 
                      2)\) \((m - 1)\)/\((5 \((m + 1)\) m)\))\) \((\(w[0, 
                      1]\)[z])\)^2*\(w[0, 2]\)[
                z]\ 
		 + \ \((\((5  m + 
                      2)\) \((m - 1)\)^2/\((5 \((m + 1)\) m^2)\))\) \((\(w[0, 
                      1]\)[z])\)^4\ 
		 + \ \((\((m - 1)\)/3)\)*
              D[\(w[1, 2]\)[z], z]\ 
		 - \ \((\((m - 1)\)^2/30)\)*
              D[\(w[1, 1]\)[z], {z, 
                  2}]\ 
		 - \((\((m - 1)\)^2/\((3  m)\))\)*\(w[0, 1]\)[z]*
              D[\(w[1, 1]\)[z], 
                z]\ 
		 - \ \((\((5  m - 
                      4)\) \((m - 1)\)^2/\((15 \((m + 1)\) m)\))\)*
              D[\(w[0, 1]\)[z], z]*\(w[1, 1]\)[
                z]\ 
		 - \ \((\((m - 1)\)/3)\)*\(w[0, 1]\)[z]*
              D[\(w[0, 2]\)[z], z]\ 
		 - \ \((\((m - 1)\)/3)\)*
              D[\(w[0, 1]\)[z], z]*\(w[0, 2]\)[
                z]\ 
		 + \ \((\((m - 1)\)^2/15)\)*\(w[0, 1]\)[z]*
              D[\(w[0, 1]\)[z], {z, 2}]\ 
		 + \ \((\((m - 1)\)^2/
                  15)\)*\((D[\(w[0, 1]\)[z], 
                    z])\)^2\ 
		 + \ \((\((5  m + 
                      2)\) \((m - 1)\)^2/\((5 \((m + 1)\) m)\))\)*\((\(w[0, 
                      1]\)[z])\)^2*D[\(w[0, 1]\)[z], z]\ )\);\)\)

\(\(\(x[1, 2]\)[
          z_]\  = \ \((\ \ \(Invar[m, 1, 2]\)[
              z]\  + \ \((\((m - 1)\)/m)\)*\(w[0, 1]\)[z]*\(w[1, 1]\)[
                z]\ 
		 + \ \(w[0, 1]\)[z]*\(w[0, 2]\)[
                z]\ 
		 - \ \((\((m - 1)\)/
                  m)\) \((\(w[0, 1]\)[z])\)^3
		 + \ \((\((m - 1)\)/4)\)*
              D[\(w[1, 1]\)[z], z]
			 - \((\((m - 1)\)/2)\) \(w[0, 1]\)[
                z]*D[\(w[0, 1]\)[z], z]\ )\);\)\)

x[1, 1][
          z_]  =  Invar[m, 1, 1][
            z]  +  (w[0, 1][z])^2;

Clear[a]

\(\(\(x[0, 2]\)[z_]\  = \ \((\ 
          Binomial[m + 1, 3]*\(a[2]\)[z]
		 + \ \((\((m - 1)\)/2)\)*
              D[\(w[0, 1]\)[z], 
                z]
			 + \ \((\((m - 1)\)/\((2  m)\))\) \((\(w[0, 1]\)[
                    z])\)^2\ )\);\)\)

\(reduceInv[n_, i_]\  := \ 
      Module[{}, \ \ \ 
		\ \ B[n, i, 0]\  = \ Inv[n, i]; 
		\ \ Do[\ 
          B[n, i, k]\  = \ 
            Expand[\ B[n, i, 
                  k - 1]\  /. \ 
					\ \ \ \ \ D[\ \(w[2, 2]\)[z], {z, 
                      7 - k}] -> 
                  D[\ \(x[2, 2]\)[z], \ {z, 7 - k}\ ]\ ], \ {k, 1, 
            7}]; 
		\ Do[\ 
          B[n, i, 7 + k]\  = \ 
            Expand[\ B[n, i, 
                  7 + k - 
                    1]\  /. 
						\ \ \ D[\ \(w[1, 2]\)[z], \ {z, 
                      9 - k}] -> D[\ \(x[1, 2]\)[z], \ {z, 9 - k}]\ ], \ {k, 
            1, 9}]; 
		\ Do[\ 
          B[n, i, 16 + k]\  = \ 
            Expand[\ B[n, i, 
                  16 + k - 
                    1]\  /. 
						\ \ \ D[\ \(w[1, 1]\)[z], {z, 
                      13 - k}] -> 
                  D[\ \(x[1, 1]\)[z], \ {z, 13 - k}]\ ], \ {k, 1, 
            13}]; 
		Do[\ 
          B[n, i, 29 + k]\  = \ 
            Expand[\ B[n, i, 
                  29 + k - 
                    1]\  /. \ 
					\ \ \ \ \ D[\ \(w[0, 2]\)[z], \ {z, 
                      13 - k}] -> 
                  D[\ \(x[0, 2]\)[z], \ {z, 13 - k}]\ ], \ {k, 1, 
            13}]; 
		\ \ \ \ \ \ A[n, i]\  = \ 
          B[n, i, 42]; 
Print["A[", n, \ ", ", \ 
          i, \ "] = ", \ A[n, i]]\ ]\)

Do[ reduceInv[s, i] ,   {s, 2, 8},  {i, 1, n[s]} ]

F[2, 1]  =  Invar[2, 1, 1][z];

F[3, 1]  =  Invar[2, 1, 2][z];

F[4, 1]  =  Invar[2, 1, 1][z]^2;

F[4, 2]  =  Invar[2, 2, 2][z];

F[5, 
          1]  =  Invar[2, 1, 1][z]*Invar[2, 1, 2][z];

F[6, 1]  =  Invar[2, 1, 1][z]^3;

F[6, 2]  =  Invar[2, 1, 2][z]^2;

F[6, 
          3]  =  Invar[2, 1, 1][z]*Invar[2, 2, 2][z];

F[6, 
          4]  =  ( Invar[2, 1, 1][z]*
              D[Invar[2, 1, 2][z], z] - (3/2) 
              D[Invar[2, 1, 1][z], z]*Invar[2, 1, 2][
                z] );

F[6, 
          5]  =  ( Invar[2, 1, 1][z]*
              D[Invar[2, 1, 1][z], {z, 2}] - (5/4) 
              D[Invar[2, 1, 1][z], z]^2 - 
            4  a[2][z]*Invar[2, 1, 1][z]^2 );

F[7, 
          1]  =  Invar[2, 1, 1][z]^2*Invar[2, 1, 2][
            z];

F[7, 
          2]  =  Invar[2, 1, 2][z]*Invar[2, 2, 2][z];

F[7, 3]  =   Invar[2, 1, 1][z]*D[Invar[2, 2, 2][z], z] - 
          2  D[Invar[2, 1, 1][z], z]*Invar[2, 2, 2][z];

\(\(F[7, 
          4]\  = \ \((\ \(Invar[2, 1, 1]\)[z]*
              D[\(Invar[2, 1, 2]\)[z], {z, 2}]
		 - \((7/2)\) 
              D[\(Invar[2, 1, 1]\)[z], z]*
              D[\(Invar[2, 1, 2]\)[z], z] + \((21/10)\) 
              D[\(Invar[2, 1, 1]\)[z], {z, 2}]*\(Invar[2, 1, 2]\)[
                z] - \((72/5)\) \(a[2]\)[z]*\(Invar[2, 1, 1]\)[
                z]*\(Invar[2, 1, 2]\)[z]\ )\);\)\)

F[8, 1]  =  Invar[2, 1, 1][z]^4;

F[8, 
          2]  =  Invar[2, 1, 1][
            z]*Invar[2, 1, 2][z]^2;

F[8, 
          3]  =  Invar[2, 1, 1][z]^2*Invar[2, 2, 2][
            z];

F[8, 4]  =  Invar[2, 2, 2][z]^2;

F[8, 5]  =  Invar[2, 1, 1][z]*F[6, 4];

F[8, 6]  =  Invar[2, 1, 1][z]*F[6, 5];

F[8, 
          7]  =  ( Invar[2, 1, 2][z]*
              D[Invar[2, 2, 2][z], z] - (4/3) 
              D[Invar[2, 1, 2][z], z]*Invar[2, 2, 2][
                z] );

F[8, 
          8]  =  ( Invar[2, 1, 2][z]*
              D[Invar[2, 1, 2][z], {z, 2}] - (7/6) 
              D[Invar[2, 1, 2][z], z]^2 - 
            6  a[2][z]*Invar[2, 1, 2][z]^2) ;

F[8, 
          9]  =  ( Invar[2, 1, 1][z]*
              D[Invar[2, 2, 2][z], {z, 2}] - (9/2) 
              D[Invar[2, 1, 1][z], z]*
              D[Invar[2, 2, 2][z], z] + (18/5) 
              D[Invar[2, 1, 1][z], {z, 2}]*Invar[2, 2, 2][
                z] - (112/5) a[2][z]*Invar[2, 1, 1][
                z]*Invar[2, 2, 2][z] );

F[8, 
          10]  =  ( Invar[2, 1, 1][z]*
              D[Invar[2, 1, 2][z], {z, 3}] - 
            6  D[Invar[2, 1, 1][z], z]*
              D[Invar[2, 1, 2][z], {z, 2}] + (42/5) 
              D[Invar[2, 1, 1][z], {z, 2}]*
              D[Invar[2, 1, 2][z], z] - (14/5) 
              D[Invar[2, 1, 1][z], {z, 3}]*Invar[2, 1, 2][
                z] - (268/5) a[2][z]*Invar[2, 1, 1][z]*
              D[Invar[2, 1, 2][z], z] + (376/5) a[2][z]*
              D[Invar[2, 1, 1][z], z]*Invar[2, 1, 2][z] + (26/5) 
              D[a[2][z], z]*Invar[2, 1, 1][z]*Invar[2, 1, 2][
                z] );

F[8, 
          11]  =  ( Invar[2, 1, 1][z]*
              D[Invar[2, 1, 1][z], {z, 4}] - 
            7  D[Invar[2, 1, 1][z], z]*
              D[Invar[2, 1, 1][z], {z, 3}] + (63/10) 
              D[Invar[2, 1, 1][z], {z, 2}]^2 - (412/5) a[2][
                z]*Invar[2, 1, 1][z]*D[Invar[2, 1, 1][z], {z, 2}] + 
            98  a[2][z]*D[Invar[2, 1, 1][z], z]^2 + 
            10  D[a[2][z], z]*Invar[2, 1, 1][z]*
              D[Invar[2, 1, 1][z], z] - 
            4  D[a[2][z], {z, 2}]*Invar[2, 1, 1][z]^2 +  (864/
                  5) a[2][z]^2*Invar[2, 1, 1][
                  z]^2 );

Expand[ A[2, 1]  -  F[2, 1] ]

Expand[ A[3, 1]  -  F[3, 1] ]

Expand[ A[4, 1]  -  F[4, 1] ]

Expand[ A[4, 2]  -  F[4, 2] ]

Expand[ A[5, 1]  -  F[5, 1] ]

Expand[ A[6, 1]  -  F[6, 1] ]

Expand[ A[6, 2]  -  F[6, 5] ]

Expand[ A[6, 3]  -  F[6, 2] ]

Expand[ 
      A[6, 4]  -  ( F[6, 4]   +  (1/4) F[6, 5] ) ]

Expand[ 
      A[6, 5]  -  (  F[6, 3]  + (1/3) F[6, 4]  + (1/20) 
              F[6, 5] ) ]

Expand[ A[7, 1]  -  F[7, 1] ]

Expand[ A[7, 2]  -  F[7, 4] ]

Expand[ A[7, 3]  -  F[7, 2] ]

Expand[ 
      A[7, 4]  -  ( F[7, 3]  +  (1/3) F[7, 4] ) ]

Expand[ A[8, 1]  -  F[8, 1] ]

Expand[ A[8, 2]  -  F[8, 6] ]

Expand[ A[8, 3]  -  F[8, 11] ]

Expand[ A[8, 4]  -  F[8, 2] ]

Expand[ 
      A[8, 5]  -  ( 
          F[8, 5]  +  (1/4) F[8, 6] ) ]

Expand[ A[8, 6]  -  F[8, 8] ]

Expand[ 
      A[8, 7]  -  ( F[8, 10]  +  (1/4) F[8, 11] ) ]

Expand[ 
        A[8, 8]  -  (1/20) F[8, 6]  - F[8, 3] - (1/3)*F[8, 5]] // 
      Factor

Expand[ A[8, 9]  -  F[8, 4] ]

Expand[ 
      A[8, 10]  -  (  F[8, 7]  + (1/3) F[8, 8] ) ]

Expand[ 
      A[8, 11]  -  ( 
          F[8, 9] + (1/3) F[8, 10]  +  (1/20) 
              F[8, 11] )]