Just to put my two cents in. Newer transmissions use or should be using a synthetic oil in them which should let you run 220 with no issue.
Just as a example.
My 2006 chevy colorado with a 4 speed transmission 4l60e. The factory computer will put the truck into a limp mode at transmission temps of 270*f. I installed a aftermarket B&M stacked plate cooler.
I tow in 3rd gear as Gm has told me to do. When towing up big hills with the a/c running my transmission temp will be about 194*f. And normal running around town in 4th gear no tralier I am about 149*f to 167*f depending on outside temps. These readings are from my edge insite which is plugged into the obd port.
Sorry if I missed it but what are your transmission temp showing.