Legally speaking, I don't know how the C7 can expect to quit the league and then take its name with them. That just won't fly.
Their alternative is to vote out the other schools, while they still have a majority.
Then it becomes a breach of contract case where the C7 are paying damages to the ones they kicked out (or if you prefer, those whose offers were rescinded).
It seems to me the only way they keep the NCAA credits, conference name and Big East contract with MSG is to remain in the conference and kick the others out. That would take some serious balls.