Take the truck to a scale. Sometimes they are at moving van company lots, and they can check the weight for about $10. Some larger truck stops have them called CAT scale, also about $10. Check the GVWR for your truck on the inside of the drivers door. Subtract the curb weight from the GVWR, and see how much weight you can carry.
If the camper is about 1,200 pounds empty, it might work, but if it has a 20 gallon fresh wate tank (160 pounds) a refrigerator that will hold 50 pounds or so of food, and your clothes, bedding, and passengers all add up to about 500 pounds more stuff. That will increase the curb weight by about 2,000 pounds. If your GVWR is say 7,700 pounds and the curb weight is less than 5,500 pounds, it will work fine. But I don't think it will work very well. You will have to check it and see.
I had a F-350 with a 460" engine in it, single rear wheels and 9,600 pound GVWR. It was fine, andsettled down 4" when the 2,200 pound camper was on the back. It did not sway excessively in the turns, and still had the capacity to tow a trailer with a light dune buggy on it. Your results should vary.
Fred.