I would like to point out that we don't have even classical way of proving something to be cryptographically sound, since we don't know if really P!=NP. Perhaps there is a constructive proof of P!=NP and this will actually let you prove cryptographic strength of something constructively.

And if you are already taking P!=NP as an axiom, what difference it makes that constructivism takes out just another axiom that you assume to be true?