I know I would.
Starters are valuable. Lefthanded power starters even more so. Guys like Bedard do not grow on trees and are worth taking a few risks to keep around. I know the guy supposedly has character issues, but he's been about what you'd expect for us so far, and the upside reward if he can pitch a healthy year for us would be exciting.
Our rotation next year without Bedard has a question mark at the end, since we're not sure what we expect from Daisuke. Right now it looks like Lester, Beckett, Buchholz, Lackey, and one of Miller or Wakefield. Now Miller looked pretty good last night, and has definitely showed progress and signs of far greater consistency for us than he's ever shown before, but earmarking a regular rotation spot for him is a stretch, and as for Wakefield, his best use is as emergency depth. We could definitely make use of one more good quality starter, especially a power LHSP. And with Miller in the wings (pretty sure we keep him, there's an option in his contract IIRC and a guy who's 7-1 mid 4's ERA would be silly to let walk) we're well protected in case he spits the bit or can't get the job done.
So if you can get him to sign, I would definitely do it. He is a man who can help us going forward if he wants to stay. Just don't earmark him mentally for more than about 150 innings, and handle him with a bit of care. These are things we can do here with our depth, so I don't consider that a bar to holding on to him.