It won't be a case of Joe fighting for his spot if Bravo comes. Or at least I'm pretty sure it won't. Here's why:
Bravo plays for the best club in the world. His only reason to want to leave is that he is regularly rotating with Ter Stegen, and both want to be first choice in every competition. He's said before that he's not happy about the situation. Why would he come to a smaller club if he didn't have assurances that he'd be #1? It doesn't fit.
Neither Hart or Bravo will be happy with second choice, and I doubt either would want to rotate between competitions. There is no way in hell we're buying Bravo to replace Caballero as second choice. I think we take Bravo for two or three seasons, and by the time he's done one of Gunn and Rulli will hopefully be the keeper Pep wants.
However, if Hart is willing to be professional, work hard and adapt, then I'd much rather we stuck with him. Gave him a season to see if he can adapt to Pep's style of playing, although at the age of almost 30, it's not easy for a goalkeeper to improve their passing and distribution by as much as Pep wants Joe to. His average pass completion last season was only 52%- not good enough for Pep. By comparison, Caballero managed over 80% completion yesterday. Neuer usually manages about 90%.