I want there to be a comic book in which Damian finds a picture of Dick wearing this and is completely insufferable for the entire issue. Because he would never let him live an outfit like that down. :P

There wasn't anything about that specifically in Batwoman, but Bette was not super on top of things, either. Which admittedly might be some kind of winking continuity reference to way back when, because the very first Batgirl had her same name and was Batwoman's sidekick and was a bit incompetent.