How did the team split, anyhow? Given Wally's bitching about being yanked by a lasso I'm guessing he was on the anti-Bruce side
It was the fallout to Tower of Babel. The League discussed the matter of Batman's continued membership. J'onn, Wally, and Kyle were on the side of keeping him on the team. Diana, Eel, and Arthur wanted him gone. We don't know which way Clark voted because when they went to talk to him they found that he had gone.
Basically, it was like Shayera's departure in JL, only she had the decency to formally make her departure to their faces.
Of course, had it been me I would have simply had Diana lasso the dream catcher to expose its falsehood, but I guess we wouldn't have gotten the story.
I don't think that would have worked. The rest of the story showed that the dreamcatcher altered reality; its effects may have been based out of dreams, but that didn't make them any less true in that specific context. The dreamcatcher's true nature would have been revealed, but they already knew what it was.