The stock cooler on my RV was aparently WAY undersized.
After a C6 rebuild this year, I put one of these, about $100 shipped from RockAuto:
https://www.haydenauto.com/en/ecatalog?partdetail=779
I also put a deep aluminum pan with a temp gauge port:
Our Hi-Tek deep transmission pans are constructed from A356-T6 aluminum and are cast with thick internal and external fins, adding additional strength and increased cooling properties. The deep pan design increases fluid capacity by 2 quarts over the stock O.E. steel pan. All these features...
www.holley.com
The original cooler vs the 779:
You must be registered for see images attach
I put two temp gauges; one on the TC output line, and one in the pan.
The pan climbs as high as 210 when crossing mountains, but I stop and let it cool down before it gets there, if possible. I can only imagine how hot that thing got with the stock cooler and the stock pan, because it had no temp gauges then. It lasted until about 80k before it was totally trashed (pump, drum, everything). The pan staying at 195 isn't uncommon for me, but it's a 27ft Class C motorhome pulling a 5x8 fiberglass box trailer.
If you aren't towing, 220 seems excessive. Consider slowing down to lower the temps, at least until you put a bigger cooler. What temp is your coolant running at? The C6 uses the radiator for some cooling, so if your engine is running hot, that can bleed into the ATF temp and reduce it's cooling.