In the comments under this answer, mike stone proposed an interesting look at how this might work:
[the] question can be formulated more physically by asking what happens in finite depth square well on the entire line. Then a self-adjoint momentum operator exists (but does not commute with the Hamiltonian) and the overlaps do give the possible momenta, [and] one can then explore what happens as the depth increases.
I think that's an excellent way to look at it, so: what are the momentum properties of the finite-well eigenstates, and how do they change in the limit of a very tall well? How do they mesh with the properties of the (possible self-adjoint extensions of the) momentum operator on the infinite square well, and its interactions with the hamiltonian over the restricted-interval version?