Remember that site just lists how much a vehicle with that towing package can PULL; not what it can carry on its back. (You need to be under both!)
For that you need your individual available payload.
Your door post lists the maximum weight that can be ON the frame of your vehicle (GVWR). Go to a CAT scale and weigh your truck with your family and gear in it, and subtract that "actual un-hitched" weight from the GVWR.
That will give you your available payload to carry the camper's tongue or pin weight. To have a reasonable towing experience (truck handling and sway control), the tow vehicle must be capable of carrying 10-15% of the total loaded camper weight for travel trailers (12% optimum) and 15-25% of pin weight for 5th wheels (20% optimum). (My 5th wheel sits at 16%)
For a travel trailer, when you find the available remaining payload, divide that weight by 0.12 to find your optimum camper weight for YOUR truck. (For example, if your remaining payload after your family and gear in it is 800 pounds, the maximum camper weight you should be looking at is 800/0.12 = 6,600 pounds regardless of the listed max tow rating - which should be a lot higher.)
For a 5th wheel trailer, when you find the available remaining payload, divide that weight by 0.2 to find your optimum camper weight for YOUR truck. (For example, if your remaining payload after your family and gear in it is 1400 pounds, the maximum camper weight you should be looking at is 1400/0.2 = 7,000 pounds regardless of the listed max tow rating - which should be a lot higher.) (In my case my max tow is 15,400 pounds but since my available payload unhitched is 1500 pounds, my 9,200 pound camper (loaded at 16%) is right in there.)
Had my camper's pin load come in any heavier, say at 20%, my pin load would be 1840 pounds and my truck would be overloaded. At 20% my maximum camper weight would have been 7500 pounds.
So hit the cat scale with your truck and happy search for your perfect camper!
www.catscale.com