The camper sticker does not take into account the weight of anything you put in the camper, like those chairs, the grill, food, water, clothing, beer, bedding, etc..
The door sticker includes PASSENGERS too.
If you're only 60-70lbs apart between the door sticker and the empty camper, you're really not too bad off with a 3500HD pickup. You'll probably only end up 750-1000lbs overweight, but still within the limits of your truck's axle ratings.
TECHNICALLY, you are supposed to keep your payload at or below the number on the door sticker, but there is no law against exceeding the factory ratings on a pickup truck in the United States that I'm aware of, or that anyone has been able to produce. The truck also has individual axle ratings, which when added together, exceed the GVWR (the number which the door sticker payload capacity is based off) by around 1500lbs on a 2500HD pickup, and as much as 3000lbs on a 3500HD dually. Most of that "extra" capacity is in the rear axle, where your camper will ride.
It is quite common for TC owners to ignore the GVWR and factory payload rating of their pickups and load it based on individual axle capacities. Some go so far as to upgrade tires and springs and load their trucks beyond even the factory axle ratings.
Please note that this is a PERSONAL choice, and you accept all liability and responsibility should there be a problem with it. As long as you drive carefully and keep your vehicle maintained properly, your risk of problems is very low.