I'm certain you'll have more than 250 pounds of stuff in the camper so you're too low there.
I had a Chevy SRW 3500 carrying an 11' camper. Total weight of about 11,200. I also used a 48" hitch extension to tow my 20' boat on a tandem axle trailer. Boat and trailer weighed about 4,500. I also had very light tongue weight, about 200 pounds. This is completely normal and almost unavoidable with a boat as ALL the weight is in the back. The boat towed great. I had no sway at all.
The biggest problem I had was tires. I was carrying nearly 7,500 on the rear axle. I think the tires were rated 3,750 each. I would have upgraded to 19.5" tires and wheels but instead just upgraded to a dually.
I really think you will need (or want) 19.5" wheel and tires. Otherwise, although quite heavy, I think it should be fine. Mine combo was.
Good luck with the 6.0. I hope it does better for you than the 2 disasters we had at work.