Yes Batman could exist.
In the movies and comic books, the police liaise with him. Batman is basically an extension of the police force. The cops even set up a Bat Signal on the rooftop. He is effectively a policeman. The cops just use plausible deniability to deny any direct link. I imagine the cops caught him several times, but always let him go (and probably even help him).
There would never be any serious manhunt for the guy. In fact, if Batman was ever killed in action, the cops would probably suit up "the next guy" and send him out there as Batman, and then deny he was ever caught and killed.
But what a lot of people are missing when they claim Batman couldn't exist is that Batman actually has the (secret) consent of the police.
What Batman actually is, is a secret police. He is the Gestapo or the SS. He does the really nasty stuff.