Get the overload springs doing some work before making the sway bar determination. You're spot on there. Try with less pressure in bags to simulate installing stable loads. Sitting on the overloads will help sway a bunch. Sway bar will help even more but one thing at a time.
Stiffer the tires the better. 16s really limit you on tire selection in the heavy weight ranges. Imo the easy and cheapest solution here is find some newer takeoff wheel/tires. For less than the cost of new tires alone you can get some new OE Ford 18s or 20s with like 3750lb cap tires. Doesn't hel the snow ability as takeoffs are typically hi way tires and honestly you got the wrong truck for snow but with camper weight and different siped and or studded tires and chains you can have ok perf in the snow. You will kill good snow and ice tires quickly in the summer with the camper though. Everything's a trade off, right.
Otherwise looks like a tight rig! Have fun w it!
Edit. just seen its a 4x4. Doesn't change any of the other reccomendations other than it will be good in the snow. Highway OE type tires still typically s uck in the snow especially wet greasy snow.