I think there is no difference between constructive and classical logic here? The proof of "the powerset of the natural is uncountable" (or equivalently, expanding out the definition of uncountability, "there does not exist a bijection between N and P(N)") is constructive, and will hold just as well in intuinistic logic. You can't get away from uncountable sets.
On the other hand, you could prove a metatheoretic result, constructing a model which only contains sets which are forced to exist by some finite formula (and therefore the model is countable). But that you can do in ordinary classical set theory also (the Löwenheim-Skolem theorem, Gödel's constructible universe).
On the other hand, you could prove a metatheoretic result, constructing a model which only contains sets which are forced to exist by some finite formula (and therefore the model is countable). But that you can do in ordinary classical set theory also (the Löwenheim-Skolem theorem, Gödel's constructible universe).
I guess this is analogous to how, in constructive logic every function is computable, but it is not possible to internalize this fact as an axiom in the logic. (https://existentialtype.wordpress.com/2012/08/09/churchs-law...)