Well, every writer sees the lasso differently. The party line since Byrne has pretty much been "it's just a focus for her inherent Truthiness," but if Gail wants to mess with that there's certainly no iron-clad "Bruce doesn't kill" level of canon to stop her.
How did the team split, anyhow?
Last time around I posted the Babel arc before League of One, which I probably should have done this time too, just for context. Oh well. Basically Bruce had Contingency Plans to take out all the other Leaguers, which he didn't bother to tell them about; Ra's got his hands on said plans and used them against the League. In the aftermath, everyone was seriously pissed off at Bruce and they voted him off the satellite, four to three. J'onn, Wally, and Kyle voted for him to stay; Diana, Arthur, Plas and Clark wanted him gone. The League then split along those lines and were pissy at each other until he came back.
I would have simply had Diana lasso the dream catcher to expose its falsehood
Well, the dreamcatcher's not exactly a falsehood. One's astral self is, on the astral plane, as legit an entity as one's normal self in the real world.
The problem is, as we've discussed before, that Batman has to be the irreplacable tactical genius for him to justify his presence in the League, operating on a plane beyond even the champion of Athena.
See, most of the time that works, but he wasn't even there for that one. Surely there's no need to downplay her talents when doing so doesn't even do anything to boost up anyone else.