I'm wondering if you have an air flow issue. The factory cooler should work just fine. That trailer is well within the capability of the truck.
I'm curious, is the grill shell the factory one, or is it an aftermarket replacement? Some aftermarket grills restrict air flow and cause transmission overheating. Or is there something else behind the grill?
I had an aftermarket grill shell on my old '08 Sierra, it was on when I bought it. It always ran close to 200 towing, which didn't really bother me. Until towed on the bypass around Indy on a 98 degree day. It was running 215-230. I realised it was the grill, took it off for the rest of the trip and it never ran higher than 190, and that was in the mountains in TN. Most of the time it was at 170-180.