I doubt McCain would accept. In TTL, McCain would probably being eying the Republican nomination for 2004.
I think he might if Lieberman casts it correctly -- tells him the role he envisions, tells why he needs McCain's help, says what he wants to accomplish.
And, though the alternate party VP thing doesn't happen IOTL, it certainly wouldn't preclude McCain from running. After serving a few years as VP under Lieberman, he could say, "I've been honored by the President's trust in me and I'm proud of what we've achieved together. He's a smart man, with many good ideas. However, his policies and cabinet still tend to run left of what's best for our nation and I feel we need to adjust our vector. Therefore, I declare my candidacy for President of the United States. I will continue to serve President Lieberman until his term is up, unless he feels the election makes that impossible. Further, I promise that, in my administration, I will give every consideration to good people from the Democratic Party, or others, for VP and my cabinet."
If he did that and they then kept the election civil while continuing to work together, that'd be an amazing example of bipartisanship and working together.
As far as Lieberman's cabinet, he wouldn't fire everyone, but he could make it clear that McCain is part of the team. Then, as some cabinet members eventually resign (or get fired), he could appoint more Republicans (or Libertarians) if he wished.