My FW UVW is 8100 and dry pin weight of 1170lbs. It weighs 9500lbs loaded, ready to camp and the pin weight is 1500lbs empty tanks.
My 2011 F150 GVWR is 7700 with 2080Lb payload. Your payload is 1660lbs. I'm 200lbs over the 4000Lb rear axle rating (5%). I also upgraded to LT tires.
It tows and stops great, no problem with hills. I have not towed in true mountains and probably never will. Everything is great UNTIL I fill the black and gray tanks. The tanks are forward of the FW axles, putting a lot of extra weight on my trucks rear axle, and I can tell the truck is not happy with it. My guess is with the tanks full I probably have at least 4600lbs on the rear axle.
From my experience and considering the information you provided, I give your idea a thumbs down. The published 1325lb pin weight is unloaded (dry). That will go up by 100's of lbs. You will run out of rear axle payload before you even get in the truck. Your truck can TOW this camper, it just can't CARRY it well.