Being able to carry the tongue weight without a WDH, assuming its in the proper percentage of total trailer weight, is a different animal than sway control. Unless the WDH has integral sway control, you'll probably have sway with P tires. LT tires will help the truck keep its rear in line but if the trailer wants to shove the hitch sideways, you'll still feel it. That's what a sway control is for - to limit the sideways shove at the hitch.
I used to tow a 4,000 lb TT with a WDH and no sway control (had it - never needed to hook it up). Of course, the TV was a GMC 2500HD diesel with LRE LT tires (planned for a future 5er, which we now have).
2015 Rockwood Signature UltraLite 8282WS Platinum, GY Marathon LRD, TST 507RV TPMS
2005 GMC 2500HD CCSB D/A, Curt E16, Prodigy P2, Garmin RV760LMT w/BC-20 b/u cam