Comic book facts aren't set in stone, so why make a big deal out of it? Batman's rules have shifted in movies, shows, and books for years. Would your coworker even care about the original comics, or just the version they know? Maybe bring up a cool story where Batman faced this choice, instead of just saying they're wrong. Keeping the peace at work might be better than winning a nerd debate.
Yeah, park.miles has a point about comic facts not being set in stone. I mean, I was in a similar spot where a friend insisted Batman never kills. Instead of just saying he's wrong, I brought up that time in Batman v Superman where he clearly uses guns on cars. Idk, it led to a fun talk about how different writers handle his rules. Maybe it's just me, but focusing on the cool stories works better than arguing. Keeping it light helped us both learn something without getting annoyed.
Batman's first appearance in 1939 had him using a handgun. The golden age stories are full of him shooting villains or letting them die. I get the idea of keeping talks light, but sometimes a fact is just a fact. If someone's confidently wrong about a core part of the character's history, correcting them isn't being a jerk. Isn't it better to actually know the material instead of just going with any version someone likes?