And then he'd meet Steph, and there is no way she wouldn't end up in a Robin or Batgirl suit.
BB!Bruce: "So. What's your story?" Steph: "Umm...somehow, I don't think you want to know."
Well, not his. BB!Bruce is eons beyond shame. Should the Birds hear about it, though, there'd be cackling at Babs.
Oh, definitely. The awkwardness would mostly be felt by DCU's Bruce, Babs and Dick. Ahh! And then what if you had BB!Bruce being all angsty and angry over Babs being paralysed by the Joker?!? That would be darn right explosive.