Really like that one. My first reaction was "and yet the Gatekeeper can still say no and kill you". After all, Voldemort's trying to prevent untold destruction, a prophecy whose exact paths to possible fulfilment are a mystery. Killing a limited number of Dementors is less important.

But my understanding of the AI box experiment is that it was never just about finding an argument that will look persuasive to someone armchair-thinking about it. It's about finding an opening to the psyche, an emotional vulnerability specific to your current target. Voldemort doesn't seem to have a lot of those, but we do have this:

Harry asked his dark side what it thought of death.

And Harry's Patronus wavered, dimmed, almost went out upon the instant, for that desperate, sobbing, screaming terror, an unutterable fear that would do anything not to die, throw everything aside not to die, that couldn't think straight or feel straight in the presence of that absolute horror, that couldn't look into the abyss of nonexistence any more than it could have stared straight into the Sun, a blind terrified thing that only wanted to find a dark corner and hide and not have to think about it any more -

So yeah, that might work.

My second objection was that if Dementors are considered national weapons in case of war, destroying Azkaban would weaken the country Voldemort intends to rule. Obvious solution, if Voldemort brings this up: kill some other countries' Dementors.

Perhaps one thing I'd change is not tell about Hermione being the solution until Voldemort agrees to do this and to revoke his threat to Harry's parents and friends, only promise to say what the solution is once they're about to attack Dementors.

EDIT: Bonus points if Harry manages to say something Quirrellike-cynical about how he had asked Dumbledore to come with him to kill Dementors, and he said no, and it took Voldemort to say yes.