IMO, it's not the towing capacity that you'll exceed but more likely the payload or carrying capacity that you'll exceed especially
on a Dodge (coming from a Dodge owner).
What does your sticker say for the Tire and Vehicle Loading placard?
The part where it says "The combined weight of occupants and cargo should never exceed XXX lbs."?
IMO, a 7,700-8,400 lbs. fifth wheel is very likely going to have a minimum of 1,540-1,680 lbs. of pin weight. And, it'll only go up as you put stuff into the camper.
If you exceed the truck's GVWR, then next thing you need to look at is the tire loading capacity and rear axle weight rating.