> SAT solvers generally have poor support for constructive logic, the kind of logic used in theorem provers based on dependent types like Coq.

Arguably, this is no longer the case. FStar [1] has dependent types, monadic effects, refinement types and a weakest precondition calculus (i.e. a richer type system than Coq) and uses the Z3 SMT solver to discharge many proof obligations. I've been using FStar recently and it works surprisingly well.

[1] - https://www.fstar-lang.org/