What ever truck you look at, you need to calculate your entire payload requirement, and compare that to the payload of the individual truck. Depending on installed options, each truck will have it's own available payload. There are F150's with as little as 750 lbs available payload, and, there are F150's with as much as 3100 lbs available payload. Many versions of Tundra, have 12 - 1300 lbs available payload. Ram and GM trucks, also have variations in their carrying capacity.
Check the tire and loading sticker (on drivers door post) on every truck you look at.
Average camping load (dishes, pots and pans, camping gear, BBQ, groceries, and water, will add 800 - 1000 lbs to the trailer's dry weight.
The trailer you listed above, when loaded for camping, will need close to 1000 lbs of available payload.
That means:
Add up the weight of all the aftermarket accessories (bed liner, bed cap, running boards, etc), people, and cargo, you plan to haul, plus 100 lbs for a weight distributing hitch, and 12 - 15 percent of your estimated (loaded) trailer weight. All of these things eat into the truck's payload.
If the truck only has 1300 lbs payload (AKA cargo carrying capacity), and you add a 200 lb bed cap, add 100 lbs of tools and camping gear in the truck bed, then hang 1000 lbs of tongue weight on the back, you would have no payload left to carry any people.
In many cases, you'll run out of payload, before you get close that mythical towing capacity number.
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