As you have determined, what the F150 can pull is significantly different from what the truck can carry.
Do consider what the travel trailer weight will be at camping trip departure.
An example & my experience:
Tow vehicle 2022 F150 3.5L XLT 145 inch WB. GVWR 7050# payload 1822#(truck tare weight is 5,228#)
2024 Flagstaff Micro Lite 22FBS (same as Mini Lite 2205s) GVWR 6794#
At the campground, Shureline tonguescale weight, fresh water one third full, grey & black empty, tongue weight: 1000# (tongue weight 14.7% of GVWR; Micro / Mini Lite trailers are tongue heavy).
Sample cargo load Calculation:
tongue weight 1000# plus 100# (for theWeight Distribution Hitch, the truck carries this) plus 350# (two people) + 125# (owner added truck accessories: ie tonneau cover, bedmatt) + 150# (camping cargo, tools in truck bed) + 35# (stuff in cab,glove box, storage box, floor mats, etc)] = 1660 # cargo weight.
So: given a “ 2013 F150 limited with3.73 gearing in the rear axle and has the 3.5 EcoBoost. The GVWR says7200 LB on the sticker and the combined weight on the yellow stickersays 1169 lbs”; your truck tare weight is 6031#. All the options(mainly electric motors and glass) add up and reduce 'steal' cargo capacity.
Using the above Sample Calculation and going 'backwards', as an example: 350# people plus 125 accessories plus 150# cargo plus 35# cab stuff = 660# minus the 1169# yellow sticker number =509# maximum tongue weight
So, a “ 2013 F150 limited ... yellowsticker says 1169 lbs”: can carry most single axle trailers with a GVWR under 4,000#, Example of a Travel trailer that the 2013 Limited is capable of carrying:
FlagstaffE-Pro E15TB GVWR 3907# will have a camp ready tongue weight of roughly 507#
So, Pulling is not the issue.
Braking, turning, driving in heavy traffic and or windy conditions, in other words, CONTROL is the issue.
'bikendan' is right on the 'money'
Good Luck in your search. Let us know what you decide. And remember a RV salesperson is trying to get a sale. And the RV salespeople I have met, never tow a trailer and do not own a tow vehicle.
Camp22FBS.