Hat-and-Cloak is Voldemort but not Quirrell. When in Quirrell, Voldemort has a whole (probably quite powerful!) brain to run his computation on. Outside of Quirrell, he relies only on what computation he can do purely as a 'ghost', or as magic, or whatever. Hat-and-Cloak is thusly disguised because Voldemort lacks a body. Or maybe Voldemort possesses someone else, who isn't as smart as Quirrell, and is proportionally dumber and more prone to mistakes. Quirrell is zombie while Voldemort's away because Voldemort set it up that way. Don't want your robot walking away without you.

Part of the groundhog-day attack involved setting up a trigger in Hermione, that when she can attack Malfoy, she should try to kill him. This explains her behaviour in the battle, and her apparent behaviour in the duel.

Hat-and-Cloak is a player in this story. Players in this story are clever and powerful. A sensible way of resolving this apparent contradiction is to postulate some form of disability or restriction applying to Hat-and-Cloak. Then all you need is Conservation of Characters.