Ok. Now I'm really confused. Using the 5th St. calculator, and using my truck's GVW (not trailer GVWR), I am ok.
Truck manufacturer specs can vary quite a bit, even across their own publications. The max 5th wheel towing capacity, Gross Compound Weight Rating, and payload specs can be quite different depending on how the truck is actually configured. Often the specs shown are best case, and unrelated to your truck.
The truck manufacturer's towing spec often shows the same number for a 2500 as for a 3500, based primarily on the engine and rear axle ratio. So it's possible your truck has a max 5th wheel tow rating of 17,400 or thereabouts. But all that starts to fall apart when payload is considered. The Gross Vehicle Compound Weight Rating (max of truck and trailer weights put together) is often lowered on the 2500s because the payload has to be considered. Fifth Wheel Safe Towing's calculator also takes payload into account and reduces the towing capacity to keep the payload at a safe level.
Some 2500s now have pretty good payload specs compared to vehicles from just a few years ago. The label inside the driver's door should tell you the payload spec on your actual truck. You can compare that to the weight you'll likely be carrying. For planning purposes, you should assume 20% of the trailer's GVWR will rest on the hitch- about 3,500 lbs. Add your 500 lbs of passengers and assume 200 lbs for the hitch, and before putting anything else in the truck, you'll be around 4,200 lbs payload. Some 2500s will have a payload spec of around 2,700 lbs. Others could approach 4,000 lbs. Your spec is on the label.