The two items are unrelated; "dropping the clutch" refers to rapid engagement of the clutch from a standstill. This hurts the entire driveline because of the sudden shock it receives. A clutch-type limitted slip differential is essentially a standard open differential that has friction disks on the spider gears. These disks allow some slippage for cornering and the like, but limit the amount of traction loss when one tire experiences less traction than the other. Typically, these differentials are rated in a percentage. Most are 25% locked, which is good for most street use. I've seen them as high as 40% locked, such as in my Alpina. Much higher than that and you're looking at a hard locker that uses gears instead of clutches.
Bottom line, on an AWD car the rear differential has little to do with easing the pain from a dropped clutch.