[TezlaKoil]

> I'm a trained mathematician as well.

Mayhap.

>A constructivist would state the result in a variety of ways. But none of them would involve a potentially self-referential construction based on the absolute truth of an infinite number of statements. Which really does rule out Cantor's argument.

But in any case you misunderstand Cantor's argument and constructive mathematics.

mbid is correct; Cantor's diagonalization argument constructively proves the uncountability of the real numbers, see e.g. Bishop-Bridges' CONSTRUCTIVE ANALYSIS, Theorem 2.19, page 29.