**Edit corrected major mistake**


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

Choose bounds for the degrees of $S,T,H$ and write them as $\sum a_m x^i y^j$ where each
$a_m$ is a fresh variable and $H$ is homogeneous. 
$H(S(x,y),T(x,y))$ is a polynomial in $x,y$ with
coefficients polynomials in $a_i$. Make a system by equating the coefficients of 
$P(x,y)=H(S(x,y),T(x,y))$
Solve the system over the rationals.

While this will work in theory, solving the system might be quite hard.
Experimenting with your example and degrees $(2,2,3)$, maple found 4 solutions in about
2 minutes.

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