Is the transmission hunting for gears during the climb? Is the torque converter unlocking to make the pull? I would probably put the shifter in manual mode and hold the trans in 4th or 5th gear (if it's a 6-speed) and make sure the converter is staying locked. Constant shifting and an unlocked converter creates a lot of heat.
How long does this hill take the climb? Really I wouldn't worry too much about the trans temp rising on a hill, as that's common. As long as it's not rising excessively or stays that hot for too terribly long. Just my two cents.
I do put the shifter in manual and make sure there isn't a lot of shifting during the climb. One of the hills is about 15 miles and an elevation gain of about 3000'. Another is about 40 miles and an elevation gain of about 3000'. The 40 mile climb seems to give me more trouble than the shorter, steeper climb. I believe it has to do with speed and duration. It's on an interstate and I try to keep the speed above 55 if I can. There is a climbing lane for a portion where I am not as concerned with speed and I dropped down to about 45 for a bit. On the 15 mile climb, I maintain speed between 30-40 mph and the temps are easier to manage.
I don't have an issue with Trans temp. Trans temp never gets above 210 since I installed the cooler. My concern was with engine coolant temps. I did read in the owner's manual that it is normal for temps to reach 250 dog when towing or climbing. This eases my mind that there isn't something internally wrong that is causing the heat. I will likely look for a larger radiator and see if that helps. Other than that, it looks like I'll just have to get used to it. I guess my expectation was not in line with what reality is, when towing. I assumed that the larger, more powerful motor would handle my trailer with ease. That isn't the case.
Thanks for the input.
Mike