I think you will find several issues here.
-calculating only 250 lb of added stuff? Are you planning to drive with no water in the camper? In my case 1200lb come from "nowhere" with only 40 gal of fresh water and empty holding tanks.
- trailers should put at least 10 percent of its weight on the tongue. Some advise 20. If you can lift tongue of 5000 lb trailer by hand, the stability will suffer.
Add a stinger to the truck and that makes good recipe for disaster.
Your and DW weight don't mean much as you'll be close to front axle.
I say you have LOT of homework to do.