RLELL

I recently purchased a 2013 Sierra HD with around 197K miles on it. I bought it to tow an enclosed trailer which weighs around 3500 to 4000 lbs fully loaded. one day while driving I switched to the transmission temp display to see what it was reading and found the temp to be running around 240 degrees. I took it in to my mechanic to have it looked at having read that anything over 200 degrees is considered to be overheating. My mechanic looked things over and told me that the transmission looks to have been replaced at some point in time and the lines also looked to be new than the rest of the underbody. The fluid looked and smelled normal. no signs of burning. So my question is... What is considered normal running temp for the transmission under normal towing load? Ive been seeing widely varied opinions but nothing that says factory-wise what is normal. If the 240 mark is high will an aux trans cooler solve the problem? Also what could be the reason(s) it is running hot? Thanks in advance