I think that Lowell would be just as good a clubhouse guy/fan favorite as Mueller currently is, and there are few players in the league that can match Mueller in that category.
I don't think Min could possibly take on Lowell's contract, and I bet Theo is currently trying to figure out what to do with Lowell if the Sox get him. If henry is open to it, the sox could get Lowell, pay part of his salary and trade him to another team (possibly Min). Otherwise, I think that Theo will try to come up with some creative move that opens a hole either at 2nd or 3rd, and will either try to convert Youk our Mueller, OR move Lowell to 1st base and get rid of Millar. With Pedroia coming up at 2nd it would be foolish to expect anyone to come in there as a long term option.
Overall, I say get Burnett if you believe you can potentially resign him for the long-term future. I know Theo doesn't always care about that, but we would be giving up quite a bit of 'pure value' to get him, by losing the small dollar Arroyo--who is, for his salary, a very effective pitcher.
On the flip side, however, Burnett is pretty damn nasty when he wants to be and the sox could use a lights out pitcher to become their new ace of the future. Burnett is at a good age for that and he and Clement would make a formidable duo for years to come.