Toy haulers are known to have heavy pin or hitch weights. That 16K lb GVWR rig will have 3250-3750 lbs pin weight once you add water and gear. It will not matter whether you are carrying the bikes or not.
Although there are ways to safely carry 4000-5000 lb loads on the rear of a F250 (I've done it for thousands of miles), I would suggest you stay with a bumper pull model instead of a fifth wheel since hitch weight will be lighter. Even the bumper pulls may require you to upgrade your current receiver from stock once you see what they weigh.