If Lieberman runs as an Independent, the first question asked will be how he will vote to organize if he wins. That will force him to align with the Democrats or the Republicans, which will make his independence moot. If he says he'll organize with the Democrats, then that will just defer the primary to the general election, and the same contest will result. If he says he will organize with the Republicans, why would the Democrats of Connecticut vote for him?
-PJ
Not really, because non-Democrats would be able to vote for him.