It seems to me that it makes more sense to approximate sinPi(x) = sin(pi x)

    scaling[x_] := 8/3  (x  - x^3)
    indices = {0, 2, 4, 6, 8, 10}
    inner[g_, h_] := NIntegrate[g h/Sqrt[1 - x^2], {x, -1, 1}, WorkingPrecision-> 40, PrecisionGoal-> 20]
    cheb[i_] := inner[ChebyshevT[i, x], Sin[Pi x]/scaling[x]]/inner[ChebyshevT[i,x], ChebyshevT[i, x]]
    coeff = cheb /@ indices
    apxi[i_] := coeff[[1 + i/2]]ChebyshevT[i, x]
    sinPi[x_] := Total[apxi /@ indices]scaling[x]
    Plot[sinPi[x], {x, -1, 1}]
    N[Simplify[sinPi[x]/(1-x^2)], 9]

This gives the following polynomial expression for sinPi(x)/(1-x²):

    3.14159264x -2.02611946x^3 +0.524036151x^5 -0.0751872634x^7 +0.00686018743x^9-0.000385937753x^11

So the function becomes:

    sub sinPi($x) {
        my $x2 = $x²;
        $x * (1 - $x2) *
        reduce * * $x2 + *,
        -0.000385937753,
         0.00686018743,
        -0.0751872634,
         0.524036151,
        -2.02611946,
         3.14159264
    }

That being said, cool stuff, definitely.