You can use Holmgren's dual method for this kind of problems. Changing variables as $\Phi=b^{-1}$, $v=b(u)$ you can rewrite the PDE as the Generalized Porous Media Equation $$ \partial_t v=\Delta \Phi(v) +f,\hspace{2cm}(\text{GPME}) $$ which has been studied intensively and for which I recommend Vazquez's book [The Porous Media Equation - mathematical theory, Oxford science publications '07]. Your regularity assumptions give here $v\in L^2H^{-1}\cap L^2H^1$ and $\Phi(v)\in L^2H^1$, which means that your solution is a weak energy solution and in particular a very weak solution (see Vazquez's book for definitions). By monotonicity of $\Phi=b^{-1}$ your statement is equivalent to showing a comparison principle $v_1\geq v_2$ for (GPME). This is well known to hold even for very weak solutions: you'll find the precise statement in [Vazquez, Theorem 6.5], which also includes the case of ordered Dirichlet boundary conditions. Vazquez's proof strongly relies on the aforementioned Holmgren's duality method.
See also this post for a related discussion.
