Sounds like a possibility.
The'02 trucks rear springs are probably not what they were 16 years ago when new. After loading the camper the old truck may need some rear suspension help. Lots of aftermarket products to choose from.
Doing the math from actual scale weights makes it easy to find out how much load capacity your truck has..... 3686 minus 1960 = 1700 lbs you'll have to work with before exceeding a tire/wheel or rear spring pack. Those small semi float rear axle on older GMs didn't like to be ran overloaded.
Also you may want to invest in a LT C or D tire to replace those OEM passenger tires.
