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.