The published hitch weight is also a dry weight, and is not realistic for real world numbers. Your trailer GVWR is 11,875. You need to multiply that number times 20% and that will be your approx. 5th wheel pin weight......2375 lbs. Now add the 5ver hitch in the bed....another 175 lbs or so, add in all your passengers, cargo, tools, BBQ grill, anything and everything that either goes in the truck or in the bed of the truck. Just the pin weight and the hitch are going to put you at 2550 lbs. Does the F150 have that kind of payload capacity.....does the F250 even have that kind of payload capacity. The F150...very doubtful.....the F250....possible, but definitely not with the 6.7 diesel, crew cab, 4x4, Lariat! To even get close to the amount of payload that you would need (probably approaching 3000 lbs or more), you could very well be in the 1 Ton SRW truck range unless you got a really stripped down F250 with a gas motor, 2 wheel drive, and an XL package (work truck).
And, as a reminder, it's never OK to go over any ONE of the weight capacity ratings....GVWR, GCWR, RAWR, FAWR, Towing capacity rating. Every single one of the truck manufacturers will clearly state to NEVER exceed any of the weight ratings. Sorry for the bad news (opinion from someone that learned the lesson the hard way), but you need a larger truck than a F150, and most likely, bigger than a F250....unless you do like I was talking about above.....gas, 2WD, etc.