the urban myth theory is great for year one stories, and ridiculous afterward. it'd be a cool lil aspect to hang on to, but it just doesn't make any sense once batman has been around for a few months, single-handedly cleaning up the city.
a much better twist would be to establish a meta-human myth about the character.
yes, everyone's gotta know by now that batman is real -- but that doesn't mean they have to know everything about him. it bothers me when batman is fighting some punk, and the punk says "i know about you, you don't kill" ... how the fuck would he know that?
there should be widely spread rumors that batman kills people. or is a vampire. or has meta human powers. or is actually an army of 60 batmen.
that keeps the "myth" aspect alive, without being (very) unrealistic.