Headcanon that Bruce uses his nicknames for his kids so much(yes everyone has their own unique nickname- he is the tired father of 8+ he has to) that the League just ...assumes that that's their vigilante names
They work with a man who dresses like a Bat who's name is Batman. Maybe creativity just isn't Gotham strong suit, who are they to judge.
Dick works fine with the League, they get away with not having to use his name that frequently, and Superman is the only one who really talks to him, so its fine that he calls him chum- the man practically raised him- until one day he's on a mission with the Lanterns and Hal screams "CHUM" at the top of his lungs and Dick just bluescreens for a second because- what?
Jason is heading out with a couple of the outlaws and passes Green Arrow who just smiles, pats him on the shoulder and goes "Have fun Firecracker. Don't hurt 'em too badly!" And Jason is like ??? Thanks??? I guess???
Tim just thinks the League is slightly condescending but also kinda soaks up all the affection because they keep calling him "sweetheart" for some reason???
Steph gets called "Treasure" a total of one time before she shuts that down real fast
Cass gets called "Princess" and is like "??? ok" and when Dick heard about it he zoomed to the WatchTower to rectify that problem immediately- she might be Bruce's Princess but she is the League's Queen, thank you very much
Damian just accepts that he is Princeling, he kinda assumes every Robin gets called that because... Batman is King??? It also leads the League to believe he and Cass are siblings- the only connection they tie from all of the Gotham vigilants
Barbara never visits anymore, but one time Bruce accidentally left her on call instead of hanging up, and suddenly seven leaguers were shouting "hey Doll!" As a greeting. She thought it was hilarious and sent it to the groupchat.
Bruce is so disappointed in his teammates. And slightly embarrassed but his kids aren't so he takes his wins.