That's like asserting the existence of a bank account in my name with a billion dollars in it that I know nothing about.

I won't ever be able to find a contradiction from that claim, because I have no way to find that bank account if it exists.

But that argument also won't convince me that the bank account exists.

That argument ought to convince you that there's a mere "possible world" where that bank account turns out to exist. Sometimes we are implicitly interested in these special-cased "possible worlds", even though they'll involve conditions that we aren't quite sure about. Non-constructive existence is nothing more than a handy way of talking about such things, compared to the constructively correct "it's not the case that the existence of X is always falsified".

It would be weird for a constructivist to be interested in a possible world that they don't believe exists.

Theoretically possible? Sure. But the kinds of questions that lead you there are generally in opposition to the kinds of principles that lead someone to prefer constructivism.