Constructive mathematicians do not assert that excluded middle is true since there is no constructive proof of (P not P). Nor do they assert that it is false, since that statement, (not (P or not P)), is constructively refutable i.e. there is a constructive proof of (not not (P or not P)). So there are no instances in constructive mathematics where excluded middle is false. But there are constructive settings where excluded middle holds, e.g. Boolean topoi, but you have to prove excluded middle is validated.
So we can't claim constructive mathematicians reject excluded middle, since they reject the rejection of excluded middle. I admit this will be confusing for people used to classical logic at first, even those for whom English is their native language.
> The way I understand it is that if you have the law of excluded middle you cannot discard 'non-constructive' proofs as invalid since they follow straight from the axioms.
This is accurate, my above point is just a pedantic one about semantics.