A number have said go to 10ply/load range E tires. Will it work or not... Not positive. I switched from 8ply/D tires on a SW 3500 to 10ply/LR E tires. Being as I only had 55-60 lbs of air in the tires, TBH, I did not feel a difference.
With above said, having just bought my first 15 series truck. XT sizes are now into play, appear to be a pseudo LR C/6 ply tire. will swag any LT style tire will be way better than a typical P metric, or the XT P metric. Tread design can also make or break towing handling too.
I'm in the "I should be able to tow trailer with out bars of any sort shape or form" group. You never know when you can loose or have one loosen up. Best to have a trailer that does not want to "fish tail" sway in its own.
Getting buffeted by side winds, trucks passing or you passing them. Should be expected. Any unexpected movement, should be side to side rocking. Or being pushed sideways as a unit. If the truly move separate, you have a problem. Find the reason, fix it!
You may have too little hitch wieght as mentioned. Too little psi in trailer truck tires for the load. Trailer axles V'd vs perpendicular to travel, not parallel. Too soft a suspension/shocks/tire sidwalls on trailer or TV. Too much wieght on one side of trailer. Hitch mounted nose up. ... I could go on. Hopefully one sees my points.
Marty