^^There is no security in Gotham.
Dent & Gordon have a nearly every scene together about how when Harvey was in Internal Affiairs he would uncover all the dirty cops in the city. Haravey accuses Gordon of not helping in which Gordon replies:"I can't afford to be an idealist, I have to work with what I'm given." It's also why Two-Face blames Gordon for why he became the way he looks now atthe end of the film because Gothams cops are corrupt and on the mobs payroll.
Plus, in "Beings" they explain Wayne Industries built Gotham City, so I would guess Bruce Wayne can enter any vacant buliding in the city and not be questioned to intensely. Notice how Gordon didn't even consider giving him a ticket for running the red light and causing an accident. The Wayne name carries weight. That's why he can walk into the building without a mask. Even if questioned he's good at playing aloof.
If Wayne Interprises helped build Gotham, then his own money would be put into any repair damage the Tumbler would cause. Putting it down as a city donation would steer away any suspicion.