Lieberman will support the rats completely if he wins. He will forget how much they wanted him gone. Because Joe is a far-left ideologue. He's not as crazy as Lamont, but Joe will side with the socialists, no matter what the socialists do to him.
Thing is, though, the rest of the Rats won't forget how he showed them and their chosen candidate up and they'll continue to backstab him. The Rats are singularly unforgiving of failure in general.
As soon as he wins he will switch back to the Rat party especially if it gives them charge of the senate.