I also think that Blackbeard's relationship with Izzy was a little codependent. That's why he's manipulated in the end. That's why - when their last interaction was being betrayed by Izzy - he returns and asks for his tea like nothing happened. Izzy gives him security.
And I think he's only able to let Izzy leave him after he has the moment in the bathtub with Stede. Where Stede makes him feel safe and like he'll take care of him.
So he's finally strong enough to let Izzy go after the duel, because he has Stede.
