Batman is willing to die for Gotham, that is made clear in Begins. He doesn't need to die to prove his point. If he can accomplish what he is set out to do, without sacrificing himself, why should he die then? I don't understand why you insist that he must die for in order to complete his character arc. Reminds me of irreconcilable MGS fanboys who wanted Snake to die in MGS4 no matter what just for the sake of poetry or whatever the fuck.
Why does he have to live? Just so the audience isn't sad at the end? I'd prefer him to die so it wasn't every other comic book movie / cartoon episode / video game.