I added a hitch to the back of my 2407T and got a hitch mounted carrier for my 300 lb WR250R. The trailer became un-towable. To much weight in the background? or too much weight taken off the hitch? I don't know but the tail was definitely wagging the Tacoma. I ended up mounting a small carrier to the hitch for the poop tank and grill, bike went in the back (angled corner to corner) but I had to add air bags. Didn't really like having the tailgate off but used four tie downs and traveled many miles without incident.
I haven't tried the bike on the back with the F150 but I expect the result would be the same. Plus the bike fits (again angled corner to corner) in the 5.5' box with the tail gate closed.