Actually, there kind of is! There’s the Batman/Superman: Authority Special #1, which is a sort of sequel to Superman & The Authority and part of Phillip Kennedy Johnson’s run on Action Comics, where Batman enlists Superman and his new Authority team, which includes Midnighter and Apollo, on a black ops mission on one of the worlds of the Multiverse trying to invade theirs. I don’t remember how much interactions they all have, but it was a fun story.
And here’s the Superman & The Authority series:
@Vroom did recently cover that story over at his Modern Superman Fan Club: