Big howdy welcome to iRV2...
I had an F350 1997 dually and it had the 4.10 axle and it could pull. I believe the f250 had the same GCWR (gross combined weight rating) of 20,000#. What will get you is the lower GVWR on the F250...I believe it is rated at 8800#,
Now you need to know what the truck weighs in full travel mode, passengers, full fuel, hitch and any cargo in the bed. This is your laden vehicel weight (LVW).
A 2 wheel drive f250, crew cab long bed will typically scale around 7200# in travel mode.
GCWR - LVW = maximum loaded trailer weight.
GVWR =LVW = maximum loaded trailer pin weight.
A 5er like you are looking at will have a pin weight once loaded that is close to 20% of the trailers GVWR. Forget the brochure dry weight ratings unless you plan to never load anything in the trailer.
So if your truck is close to 7200#...
Your max trailer weight is 20,000# - 7200# = 12800#
Your max pin weight is 8800 - 7200 = 1600#.
Based on the 20% estimate, this means that your max trailer weight will be 8000#...so you will reach your GVWR limit long before the max trailer of 12,800# is reached.
Since a travel trailer typically has a hitch weight closer to 12% of the trailer weight, you can pull a much larger travel trailer than a 5er.
Hitchhikers tend to be heavy, especially on the pin weight, so you will probably be over on pin weight and GVWR for the truck. Some people will add over load springs, air springs or such to prop up the rear axle height, but you are still over the trucks limits....so it is your choice. The truck will easily pull the trailer, provided it can haul the pin weight.