Nah, won't happen. I wouldn't give up the two picks. Briggs is a 7 time Pro Bowler and worth a first round pick by himself but we'd never get it. Even with Briggs and one third I doubt you'd do much better than to creep into the back end of the first. Briggs makes peanuts this year comparatively speaking and he's worth keeping for that alone.
If you want both a stud LT and a stud DE, you have to be willing to give up something to get something. Otherwise, the Bears are going to be stuck either giving up sacks in bunches in 2012 because of Webb or we'll still not have any applied pressure on the QB due to the weakness of the LDE. I would do it simply because Briggs is in decline and he's an asshole of a player who keeps wanting his salary bumped up every year.