One approach is to work symbolically and solve a system over the rationals.

Choose bounds for the degrees of $S,T$ and write them as $\sum a_m x^i y^j$ where each
$a_m$ is a fresh variable. Compute $H(S(x,y),T(x,y))$ - it is a polynomial in $x,y$ with
coefficients polynomials in $a_i$. Pick the degree $d_h$ of the homogeneous part and make a system
where each monomial coefficient $\ne d_h$ is zero. Solve the system over the rationals.

While this will work in theory, solving the system might be quite hard --
couldn't solve your example in 20 minutes.

Partially optimistic might be the fact that the system is overdetermined.