Re: 2008 f150 limited AWD - swap to 20" tires increase towing capacity?
There are multiple factors that determine the towing capacity of any given vehicle.
Suspension: Load rating of the springs - front and rear
Wheel base: lwb vs swb
tires: light truck (LT) vs passenger (P); load range b / c / d / e
hitch: weight distributing vs weight carrying; load rating of hitch
engine: V6 vs V8 ; HP and torque output
cooling system capacity
AND Total options: Many options add weight to the vehicle, and that increase reduces the total that can be towed. The number of passengers you carry can affect the towing capability. If it is just you and the wife, and she is 100 lbs and you are 160, that takes 260 lb of the towing capacity, If you are taking your three buddies on a camping trip, and you each weigh 200 lb, that is 800 lb of your total weight, and that weight has to be offset by 800 lbs less in the trailer. All the camping gear loaded in the bed also reduces the amount of trailer as does all the crap we throw in or on the trailer.
To answer your question however, no, just changing tires in not likely to give you that extra 3700 lb towing capacity.