This is very cool, especially the automatic proof generation!
That said, I wonder if it would make more sense to use an SMT prover as fallback rather than a SAT solver. The latter requires converting the 32-bit operations into boolean circuits, which might not be the most efficient way to solve the problem considering modern SMT solvers such as Z3 and CVC5 already have built-in efficient support for bit-vectors?
In fact, I believe that's what Alive2 [1] does, which is a similar tool (although with slightly different goals).
I think it might even be possible to make this work for 64-bit vectors with SMT (in most cases).