As an ex-CT Yankee, I can say that Lieberman is only in possible trouble in the Dem primary. He would be a candidate for a cross-endorsement (CT allows those), and a Dem win in the primary would give him a nice excuse to cross the aisle, where he would be a bit of a RINO, but still an upgrade over what CT typically offers these days.
Lieberman has a 17/100 lifetime rating from the American Conservative Union, and is worse in recent years. The idea that he would somehow make an acceptable RINO seems crazy to me.