F150 asa TV for 5th wheel

I pulled a Jayco 26.5RLS with mine for 4 yrs, 14,500 miles. Was a great combination. Many, many 6% + grades out west over the years. My F-150 has both the Max tow AND HD Payload package. Payload of 2166 lbs. New ones have a few hundred more, similarly equipped. But, I was right at max payload with just the two of us, and two dogs, and nothing else in the truck. Was also very careful to get the lightest hitch we could, and pack light. When we needed to replace due to delamination issues, all newer versions of similar fifths had gained 4-600 lbs, so I would have been way over. Had to switch to a TT. Really doubt your '02 is any lighter than my 2012 "light" was, plus the 3 extra family members...
HD Payload equipped are virtually non-existent in the used market, so you would have to order new. For that price, you could find some really, really nice gently used 250's, and maybe even 350's that can easily carry the weight.