If the dealer installed towing harness for your ATV trailer works properly, without the Cooltech harness, then probably the best way to solve this is to remove the Cooltech harness totally. Then install a separate wiring harness for the "toad" lights, using supplemental bulbs in the rear housings. Like the kit that Blue Ox sells.
2008 King Aire 4562, Spartan K3(GT) w/ Cummins ISX 600
2014 Jeep Grand Cherokee Overland 5.7L V8 Hemi w/ Blue Ox Aventa LX Tow Bar and baseplate, SMI Air Force One brake