He gives them his actual license, and they're like "Stop stealing Bruce Wayne's fucking driver's license, you weirdo! >:("
He asks for it back because he needs it to drive. They have to suffer through "I'm Bruce Wayne" in the I'm-Batman voice. (He is not getting his license back no matter what after saying "I'm Bruce Wayne" in the I'm-Batman voice.)
The next day cops are showing up at the mansion like "We regret to inform you that Batman stole your license again. Here." and give it back to him.
Bruce is like "You could at least pretend to find it plausible that I could be Batman. You don't have to be so mean about it. :("
They reassure him that they find it extremely plausible that he could be Batman, if he'll just sign for the return of his stolen property so that they can leave before he does another godawful "I'm Batman" impression that sounds more like their great-aunt Mindy who's been smoking two packs a day since The War, that would be great.
Back at the precinct, they file the 328th identity theft-related charge against Batman for pretending to be Bruce Wayne and call it a day.
When Commissioner Gordon finally keels over dead on the job of a massive stroke, it will be because of this. "You understand that, right? That what takes me out is actually going to be you and that fucking driver's license?"
Bruce is, however, very much driving the batmobile without insurance or registration because no one is going to cover a 1) rocket-propelled 2) car you built yourself 3) that you use exclusively to commit crimes, and there's no way that thing is street-legal.