Your engine should be fully capable even if you slow a bit uphill. When trailer-towing the transmission is typically the limiting factor. Adding a second transmission cooler is the right thing to do. I did that to my Dodge Ram after lunching the transmission climbing the mountain into Santa Fe while pulling a 5th wheel trailer.

The major cause of transmission overheating is usually frequent shifting under load. Manually shifting to a lower gear uphill is a good thing. On my truck, I switch the overdrive off while towing and just eat the additional fuel use.