Did you mean to say “prove there exists an element of 2^N which is not recursively enumerable” instead, or have some particular constructive system in mind more restrictive than intuitionistic logic? To be explicit, it seems to me the following serves as an intuitionistically acceptable proof that there exists a non-r.e. subset of N:

Let Halt(a, b) mean that the program coded by a halts on input b. Consider {x | ~Halt(x, x)}. If this were r.e. (i.e., semidecidable), then there would be some p such that Halt(p, x) is equivalent to ~Halt(x, x) for all x. But then, Halt(p, p) would be equivalent to ~Halt(p, p), a contradiction (even intuitionistically). Thus, we can conclude, {x | ~Halt(x, x)} is non-r.e.

The only thing which we wouldn’t be able to do intuitionistically, it seems to me, is go on to conclude that this set has a 2-valued characteristic function. I’m almost certain you just made a small typo, but just in case not, have I missed some flaw in the above ostensibly constructive proof?

The nice idea of Andrej presented in this post, was of great inspiration in writing:
http://www.mathematik.uni-muenchen.de/~chiarabi/research/Papers/Articolo%20Bauer/versione_articolo.pdf
(next CIE08). The application of such proof transformation to inductive proofs on naturals
permit to extract “tail” recursive code. Other authors investigated such
possibility, like Penny Anderson(1994), but their method were extremely much more complicated.
Here the proposed proof transformation is clear and simple. There is a point that is not yet
fully investigated: how to link such transformation to CPS? CPS is the canonical program-transformation to obtain tail recursion in programming. I think is possible to write
a CPS on proofs, that take into account also proof by inductions, acting extractly as the
proof transformation presented here. Ciao, Luca