Well, I'm not Dan, but you're towing the 310BHDS, which is a Trailer that is 35' long and weighs up to 10,000 lbs. Your round bar is a simple distribution hitch and added sway control is needed for a couple of reasons.
First, your trailer weighs much more than your truck and this contributes greatly to the potential for sway.
Second. Most people, including me, haven't experienced real sway, which is an uncontrollable occilation of the trailer that progressively gets worse. Often resulting in an accident. The push-pull from passing vehicles is not really the problem, although sway bars help to dampen that for those who experience it.
Personally, I would at least add two sway bars to your existing round bar WDH. But I would probably opt for a combination unit like the Reese Dual Cam or the Equal-i-zer.
DW, 3 Kids and our Goldens
2012 Shamrock 233S
2008 Toyota Sequoia 5.7L 4WD