Oh, Jason can be Batman. He'd be the kind of Batman that Future!Tim was in Titans of Tomorrow. He just needs some serious therapy for his PTSD and Daddy issues before that. Also, a writer who knows how the character should be written. Which is not Morrison.
On another note, it is damn scary how much alike Jason and Tim are becoming with their trauma and Daddy issues and crossing-lines-ness. Only, Tim's too hung up on what Bruce would want to kill or use guns.