I'd see a loooooooooooot of questions cropping up in that case. Hell, look at those four films they made a few years ago. Right at the start of the first of them, in the factory when he dropped the soon-to-be Joker into that vat of acid, Commisioner Gordon was trying to arrest him. And how long, really, could he keep up the charade? I mean, really...it wouldn't take the FBI long to figure out that it was someone rich dressing up as a bat and fighting crime. Then the trail would eventually lead to Mr. Wayne. "Why are you spending so much money, Mr. Wayne? What about these purchases here?" And there are numerous entrances to the Bat-Cave, I'm sure. How could there be just one in the whole place? I mean, they'd HAVE to have a back-up somewhere. Eventually, they'd find him out.
But yeah. Back on topic here...
I don't think he would be much different than the Punisher. The only difference is that he'd be more the sniper type instead of the guns-blazing type. He cloaks himself in shadows, after all, staying hidden until it's time to make his appearance. His costume would be his cover as always, but he'd be more like an assassin than anything else: discover and locate his target, and then eliminate them. Everything else would just be second...