I'm on the same wavelength as Dave. SUVs are often "tired" for a smooth ride. GM also hasn't been known for supplying overly wide rims. Your tires have a 10.3" section width and if they are on the minimum 7" rim, that is not really a good fit--8" would be better. At 32 psi I'm guessing you have a soft tire that rolls under the rim--especially at the front--when cornering. This is not the set up you want for towing. If you're satisfied with the towing, I think that is more a reflection of a large heavy tv and fairly light tt where the tail will have some difficulty wagging the dog. However, I saw some video of a swaying landscape trailer putting a 1/2 ton into the ditch, so a little tail can still wag a big dog.
I upgraded my previous Silverado 1500 to LT LRD tires and couldn't believe the improvement over OEM. I also went down to 225 which was a much better match for the 6" rims. Trial and error will help you find a better psi. I would start with 40 psi solo and 45 towing. Put all 4 at the same pressure and then decrease the front if it seems harsh. I would definitely try 50 psi on the rear for towing. You really should know the axle weights with the tv empty as well as hitched up with full camping weight on board.
I'm editing this with a little more research on tirerack.com. Your P series tires do not have load inflation tables. They are SL rated tires which means the max. rated load can be carried at 35 psi. Raising the psi to 44 or 51 does not increase the rated load, but will give better performance than carrying that same load at just 35 psi. Also P series tires, when fitted to a truck should have the max. rated load stamped on the tire reduced by 10%. This is just one link that turned up in a search at tirerack.
http://www.tirerack.com/tires/tirete...jsp?techid=21&
Henry