Vessel comes back with a counter and it will be a 0/0. Thus, it immediately dies again. It *does* trigger but the trigger goes onto the stack after it dies the second time. Since the trigger has a "if you do" clause regarding exiling it, and you can't exile it since it died, nothing happens.
Unless there is a way to boost the toughness (or just use Undying for this) Persist won't result in a Demon.
Unless there is a way to boost the toughness (or just use Undying for this) Persist won't result in a Demon.