3800 dry will quickly grow to 4500+, not to include occupants.
You are in DRW territory or, if the roads you drive don't like DRW, 19.5" Single wheels.
Your RAW will probably be around 3000-3500 lbs depending on may factors before dropping a camper in. Nearly all of the camper weigh will hit the rear axle. 6K-7Klbs is about the limit on your typical 17" and 18" LRE tires. For 17" you are closer to 6400lbs, and some 18's will get you the 7K. There are some larger options (285/75R17) that can get you to about 8K on an axle if the wheels will support it, and you want that big of a tire.
My dad's Lance has a "dry" weight of 2600lbs, but on the scale loaded for camping it's closer to adding 4K lbs to the truck.
All in all you need more than a standard LRE tire will provide, either by going with 2 extra LRE's on a DRW or going with a LRG on a 19.5 (my 19.5's are 4500lbs a tire).