Great advice on here, all with their own merit. I really do not think in real world results the F150 would vary much from the f250 in all areas including price and mpg.
Back to your original question, why the differance from the website and the dealer lots. The websites list what I call the farmers package. No air, no power anything, no carpet no options what so ever. The math your not able to do is calculations how much the AC compressor weighs, how much all the carpet, buttons, speakers, tow hooks, skid plates, 5th wheel prep packages and so on weighs. That sticker everyone is pointing you to does that, takes what the truck was built for, if it is 3000pds and starts taking away the weight of all these options.
When you are looking the items that will set these f150s with higher payload from the others on the lot, tow mirrors, crew cab or 8ft boxes. Truck would need to have those as I understand it to give you the higher payload ratings.
If I am looking at the right trailer it is 5k dry and 3k cargo, say 6500 to 7000pds on the road. Give or take 1k hitch weight and 400pds for those in the rig, see why you want more then 1350 but the trucks listed with 2k payload are plenty. Lot of the dodges age 1500pd payloads with 40mpg in the Eco diesel and the chevs have some of the best in the business right now in the 1500 line.