Ugh, you have a lot going against you. Half ton payload, large family wanting a large RV, 3.08 gear. Yuck.
For some good news. 3.08 axle was never offered with the HD towing package which includes most importantly a includes a transmission cooler. That package with the better axle ratio bumps tow rating by about 2500 pounds. Also, your 6-spd trans with 3.08 axle has very similar gearing in gears 1-4 to prior generation 4-spd trans with 4.10 axle.
Now reality. Besides tow rating, your vehicle has other limitations for RV towing. Payload at around 1500 pounds - confirm with Tire and Loading sticker on your drigver's door post. Payload should not be exceeded with the sum of people, cargo, and trailer tongue weight. Estimate the weight of your family, I'll pick a random number of 800 pounds. That would leave 700 pound for trailer tongue weight which is about 5400 loaded or 4000 pounds dry. There aren't many RVs in that weight range that are comfortable for a family of 6.
A 3-bed hybrid may be good. Make the teenagers sleep in a tent most of the time. Get that trans cooler.
