I feel you also need WDH on these heavy tongue weights because you are applying the weight behind the axle instead of on top of it like a FW. That leverage makes a big difference.
When I had my 22' toy hauler, I had one friction anti-sway bar and that was enough. Now my 20' enclosed trailer uses the same WDH but without the sway control. The TH had a taller profile and more side surface area than my enclosed which I feel makes the difference. Your wider rear track may negate the need for a sway device, but you may want one if you travel in gusty areas or where there is frequent semi traffic passing you (this is where I felt the difference in my TH).
What I did want to question is your 2000 lb tongue weight rating on your receiver. Usually only aftermarket models have that amount of capacity without running weight distribution. Are sure that your receiver was rated for that much dead weight and that was not the rating with a WDH?