Yeah I do find the "Bat" and "Cat" thing super silly. Just call each other Bruce and Selina unless someone is listening, seriously! Or just say Batman and Catwoman!
I guess every couple has their dumb names for each other so why should Bruce and Selina be any different, this when they are talking to each other at super hero gatherings they can make everyone cringe with how cutesy they are being. Even Superman will be giving them the side eye at Justice League meetings.
I still love them together, and my romantic heart just wants to see them together for a while!