There are FW you can pull with that truck, but the two you mentioned are NOT in that group. Here is the issue: Look at the driver door latch post for a yellow border placard. It will specify the max weight of all passengers and cargo shall not exceed XXXX pounds. That is your max payload, regardless of ads to the contrary.
The payload is exactly what it says. It must include the pin wt of the FW, which will be 15% - 20% of the LOADED weight of the FW, plus 200# for a FW hitch, plus the weight of all passengers, cargo, pets, tools, and anything else carried in the truck. As Steve said, the pin wt of the camper, plus a hitch will likely exceed your payload, and you are not in the truck yet!
Normally, I say 20% for a FW pin wt. TH are different because so much weight is well behind the axle, and they often will have a lower % pin wt. You need at least 15% for comfortable towing Look at a bit smaller TH FW and do your weight calculations using 17% or 18% of GVWR to estimate pin wt. Hopefully, you will find one you love.
Note, I have a F350 SRW, and my payload is 3543# My FW has a GVWR of 15,500#, and I am right at my max load.