You don't mention whether it is the Jayflight or Eagle 28BHBE. We have the 2015 Eagle and delivered with bike rack and few other options weighs in at just under 7300 lbs according to yellow sticker whereas the posted weight was 6700lbs. I pull it with a 2010 5.4 with Maxtow. The tow capacity is supposed to be 11,100lbs and I am nowhere near that. The Maxtow package adds about 300 lbs of payload and I am closer than I would like to be on payload. I do not carry full water tanks (full freshwater adds 800lbs of trailer weight and at least 100lbs of tongue weight). The 5.4 has to work pretty hard through mountains of NY, NH, VT and Maine but is able to maintain speed limit but the Eco would be better to have in the mountains.
Payload will be your limiting factor and your payload will probably be around 1400 lbs depending on the options on your truck. You definitely won't be able to take everything you want camping.
The 2015 F150's have an auxiliary oil cooler as well as additional radiator capacity and 3.73 gears in the HD Payload with Maxtow package which is required for the advertised 12,000lb towing capacity, which are likely not included on your F150.
You might be close on the rated capacities, but I think towing the 28BHBE a lot through the mountains would be a real chore with your F150 IMHO.
2015 Eagle 284BHBE
2010 Ford F150 XLT 5.4 Maxtow, Coverking Seat Covers, Putco LED interiors, Kenwood DNX571TR GPS, Kicker Powerstage, Weathertech Floor Mats and Window Vents, Line-X, Firestone Airbags, Michelin LTX-MS2, LivewireTS 5 Star tuned, Propride 3P