Sounds like the dealer is on the right track, as are many posters.
I run 10 ply LT tires on my v8 4Runner, though 6 ply is probably enough, but NOT th P tires that came with it.
Your wdh has ball angle and ball height, as well as tension arms settings.
Make sure you have MATCHING PLY tires on front and rear. Don't tow with 6 ply fronts and 10 ply rears on modern SUVs. That was fine for trucks years ago, but the common cross linked suspension in many will feel like you're balancing on a rubber ball. Ask me how I know ;)
I can tow at about any speed and it is rock steady.
It's very much worth the effort to get it all set up properly. Camping should be fun and relaxing, not a cardio stress test getting there :)