Proofs that derive a contradiction are perfectly acceptable in constructive mathematics. Indeed, the real numbers are uncountable in constructive mathematics. What is not allowed is elimination of double negatives, which is not present in the above proof.
Is there a specific issue you have with this proof? With respect, I think you may be conflating a constructive proof of the uncountability of the reals with a proof that the set of all computable reals is countable. These are two different claims.
A statement about the reals which is provable under constructive mathematics does not reduce to a statement about the computable reals and vice versa. A set can be constructively definable without all of its elements being constructively enumerable.
Given any explicit sequence of real numbers, it's always possible to generate a number not in that sequence. In fact, this can be done algorithmically using essentially Cantor's method.
