I don't think Lieberman would even accept the offer, if tendered, even though he's the closest thing they have to a real centrist.
He'd be a bad pick because he's simply not inspiring, nor does he seem like much of a leader. I'd say that he's their McCain. It would be an admission of abject failure for them to run him, after having stormed the galaxy with The One Light Worker.
Good points, and all of your analyses are solid and interesting. One factor which would throw a lot of initiative back to the Democrats however, is if the GOP runs another cull. Specifically, Mitt Romney.
If 2012 was Romney v. Rodham, with social conservatives and black Democrats deciding it was a perfect day to stay home and watch the shadows lengthen . . .
Someone would win with a plurality of the vote.