I have also cross-posted this question to MO, where it has now been answered.
Suppose we have $f : A \to B$ and $g : B \to A$, as well as left inverses $f_r : B \to A$ of $f$ and $g_r : A \to B$ of $g$. Suppose further that $f$ and $g$ have decidable images; i.e., we have $\forall b : B.(\exists a : A. f(a) = b) \vee \neg(\exists a : A. f(a) = b)$, and similarly in the other direction (replace $\vee$ with $+$ in a proof-irrelevant setting). So this is the structure necessary to work with an expression like $f^{-1}(b)$; we can ask whether it is defined, and what its value is if so.
My impression is that the inability to invert an arbitrary function is ordinarily the biggest obstacle in Cantor-Schröder-Bernstein, so I'm interested in knowing what difficulties are left if you skip over that issue by fiat.
I've been playing with these assumptions in Coq, and so far I've managed to adapt the standard proof by König to show that there exist sequences of functions $F_n : A \to B$ and $G_n : B \to A$ which gradually approximate the classically defined bijections. Most importantly, their sequences of compositions $G_n \circ F_n$ and $F_n \circ G_n$ "converge pointwise to" the identities on $A$ and $B$ in the following weak sense: $$\forall a : A. \neg\neg\exists n : \mathbb N.\forall m \ge n.G_m(F_m(a)) = a$$ $$\forall b : B. \neg\neg\exists n : \mathbb N.\forall m \ge n.F_m(G_m(b)) = b$$
But this seems like a fairly weak result, and I'm curious whether anything stronger is possible! I did find the Myhill isomorphism theorem, but having looked at the proof, it seems like any adaptation of it to more general premises like the ones I've given might require the types in question to be subcountable or something like that.
