What if McCain had decided to not let his party tell him what was and wasn't acceptable and decided to pick Joe Lieberman. Would it have made any difference in the election?
Lieberman would attract more independents
McCain is still sunk by more or less the same amount. Lieberman would attract more independents (and there would be no Palin to turn them away), but the choice of a pro-choice non-Christian would not go over well with the evangelical base, many of whom would consequently stay home on Election Day.
I like the way how you kind of just hand-waved the base acceptance. In reality, as you said at the start of the thread, the base would go nuts if McCain picked Lieberman. He would be, in effect, running against his own party - and he'd almost certainly fall on his arse in doing so.
Let's be serious here. McCain was struggling with campaign funds before he picked Palin, and his ground game was, with some exceptions, essentially non-existant. If McCain picks Lieberman, then he's running (in everything but name) a very weird, almost baseless, organisationally bereft*, third-party ticket. In effect, it's Ross Perot Mk 2.0, without any guarantee of Perot's appeal to either side of the aisle.
Unless something very weird happens, I cannot see that even being competitive with Obama, let alone winning. Whatever McCain makes up with in terms of the swing vote, he more than loses with base turnout.
*This, I should point out, against one of the most superbly organised campaigns in living memory.
Yes, I found that difficult -- but I've been reading about this recently, and I don't think there would be a mass uprising. Patrick Ruffini wrote a post, at the time, that he'd accept Lieberman, and most of his positions are conventionally conservative. Of course, he's not exactly RedState. But in light of the demonisation of Obama, would conservatives really risk an Obama victory just to stick their fingers in McCain's eyes?
Alternately if he had allowed Palin to run free it could have been a PR disaster; but it could also have worked. Palin did not do well in her first few interviews, but you could see her getting better at it; if they had marketed her as raw but learning and she kept getting better it would have energized the base even more. You look at the numbers and Ohio and Fla were lost by less then .1 percent; several other states were pretty close as well. I think Obama still wins but its a LOT closer both electorally and popular vote; probably only 30-40 electoral and maybe 3 million or so popular. Most of the popular vote edge for obama came in NY, CA and Illinois.
Even with Lieberman, there was no way McCain was going to prevent Obama from getting the Kerry States + Iowa + Nevada + Colorado + New Mexico.
And as has been amply pointed out here, Lieberman and McCain agree on the war, and that's basically it. I can't imagine anything that would make the religious right angrier than putting a pro-choice non-Christian on the ticket. Lieberman would have to either repudiate all of his views, or McCain would be saddled with a VP with whom he agreed on essentially one issue. And a single issue campaign on continuing the war in Iraq was not going to be a winner this year.