Welp, if you can find a slide in weighing in UNDER 1000lbs or so, sure you could. For your own comfort and the longevity of the truck you'd probably greatly appreciate air suspension. A 1000lbs dry camper would allow you enough extra capacity for clothing, "stuff", food, etc. Just keep in mind your tires and axles can really only support 1500lbs, and you really shouldn't plan on doing that all the time. Also remember to add in 10% of the trailer you are pulling; the tongue weight counts against your bed capacity. If your trailer is 1000lbs, your camper 1000lbs, and your stuff (including the dog) is 400lbs, you have pretty much maxed out the capacity of the truck.

Things to do:

Upgrade suspension to air suspension, ideally, or helper springs.
Upgrade tires to "D" range (I think). The next step heavier than LT tires.
Make sure your axle seals don't blow out, and if they do, replace relatively quickly.
Make sure you calculate WET weight of the camper you plan on getting. That's dry weight plus all the stuff you plan to carry.
Add 10% of the trailer weight to the weight of the camper.
Plan on a 1500lb max capacity of that truck, preferably less.

You may find ramping the bike into the truck's bed and hauling a little popup camper much more comfortable and a lot easier than you think. It'll certainly make the weight issue disapear.