For the 881 I would budget for a set of 19.5 rims and tires at around $3500. Thes will provide 8800 lbs. max load capacity at the rear axle. Payload with heavy duty trucks is mostly a function of the load capacity of all the tires at the rear axle. 6400 lbs (minus weight of the truck on the rear wheels) for the pair with factory tires, 7500 with Nitto or Toyo tires, and 8800 with 19.5 tires with a SRW truck.
With the Nitto or Toyo tires the approximate max payload is going to be 7500 - 3200 = 4300 lbs. and if you want a safer margin, a maximum load of 4000 lbs. camper and contents (best to add 1000 lbs. to camper sticker weight). That works out to a camper weight of 3,000 lbs. if you do not want to go to 19.5 rims and tires.
You will not however find 19.5 tires with a A/T tread, only with highway treads that are designed to maximize tread life and fuel economy on commercial vehicles.
At least you are getting accurate information on the weight of the camper from the dealer. The Lance published weight for an 881 is 3280 lbs. so the 3600 lb reflects what was added, like an AC or roof rack to the camper.
The 3600 lb. 881 will likely be at 4400 lbs when you leave on a trip and if you have fresh, grey, or black water fluids in the holding tanks it is easy to be carrying around an additional 160-200 lbs. and then if you decide to travel with a generator and fuel there is another 100 plus pounds to add to the truck's rear tire load.
Unfortunately you bought the truck before deciding on a camper. Lot easier to find sleeping for 4 adults with a camper built for a long bed truck. The extra 18" adds to the dinette area which becomes the secondary sleeping area.
Another used camper to consider for your new truck is the 830 with its fold out sleeping area.
There are advantages to the 855s with its slide and the metal frame construction and wiring in place for a rear camera and better insulation and dual pane windows on the 4-seasons model. The lighter the camper the more flexibility in terms of carrying gear inside or outside the camper and in being able to accommodate the tongue weight of a trailer.
Good idea to have the Lance wiring installation work done along with tie-downs installed ahead of time. Then you can travel a distance to look at a used camper and if it is what you want you can take it home with you the same day.