I'd put studs on the trailer too. To me, a tow rig with studs, and no major traction ability on the trailer, is no different than a FWD rig with studs on the front, normal street tires on the rear is icy conditions.
I have not towed at all with the hitches you are asking about. I have towed many miles in the snow, with a std WD and a Dual cam. I ran with out studs, yes legal here in Wa St. Just a traction tire on the truck, an aggressive hwy tire on the RV trailer. If in the local mtn passes, and I felt I need chains to go forward, stop safely etc, and or it was required per the WSP, I chained up the trailer along with the truck. I never had the trailer wanting to come around. That included going down a really icy 10% grade with mid 50+ MPH winds.
Not saying you need a different hitch setup. But as noted, that trailer does not have any traction compared to the truck, so it sliding sidways and in front of you, does not surprise!
marty
92 Navistar dump truck, 7.3L 7 sp, 4.33 gears with a Detroit no spin
2014 Chevy 1500 Dual cab 4x4
92 Red-e-haul 12K equipment trailer