Our operating temperature for the 4T65 is 160-220F. You can't hit 350F as limp mode comes in around 260F iirc.
Running a 4T65 as cool or cooler than 120-135F will mean that it never gets to operating temp and could end up with condensation etc not being heated out. Therefore causing water contamination inside the trans and more damage than heat.
The size of the trans cooler in the radiator end cap is pretty small. In most of the areas we drive, this should be adequate. If you live in Arizona or the mountains it would likely be a good idea to add a cooler for the hot weather.
A fresh rebuilt trans typically runs cool anyhow.
