Most 1/2 ton's don't have enough "available" payload capacity to support the loaded tongue weight of a Eagle 322FKS. Also, front kitchen TT's tend to have a heavier tongue weight then similar TT's without.
The 2011 322FKS has a GVWR of 10,250lbs, "published" dry hitch weight of 1,160lbs, and a "published" UVW of 8,875lbs......, and I bet the as-shipped UVW was around 9,200lbs.
IMO a hypothetical 2011 322FKS conservatively
loaded at 9,700lbs would put the loaded tongue weight in the 1,300lbs to 1,500lbs range with the front kitchen (BIL had one).
I would run your loaded F-150 over a CAT scale and subtract the CAT weight from your TV's GVWR......,the remainder is your "available" payload capacity.
Just food for thought.
2016 GM 2500HD 6.0L/4.10
2018 Jay Flight 24RBS
2002 GM 2500HD 6.0L/4:10 (retired)
2005 Jayco Eagle 278FBS (retired)
1999 Jayco Eagle 246FB (retired)
Reese HP Dual Cam (Strait-Line)