I may be wrong, but adding different gears will only help performance, not change the weight ratings for the vehicle. Those numbers are based on the suspension, axles, wheels, tires, etc.
Someone said that you may be expediting the death of the TV by making this sort of change for towing...I agree. Chevy designed that truck for a certain use, and overloading it will only hurt it.
Either upgrade the truck, or downsize the trailer.....though I think you may still be OK with the trailer you mentioned, but be aware of the limitations.
Did I add to the confusion:?