GM says that the stock cooling capacity is adequate for towing within the range of the rated capacities.
I've towed with an aux cooler, and without. I've only had one trans go on a tow vehicle and I bought that one with 97k on it, so I have no idea how well it was maintained. Otherwise I've towed though mountains with 2 different trucks, neither had a trans cooler. I think cooling issues while towing were a bigger problem in older vehicles. As said, monitor the temps and see if it's really an issue. The only thing I'd be concerned about is prolonged temps above 120. By prolonged I mean 20-30 minutes or more.
That said, axillary coolers are cheap and an easy DIY project. If you feel better putting one in, you don't need us telling you to do so. Just do it, as they say!
2013 Jayco Jayfeather X20 E, 2008 GMC Sierra 1500 Crew Cab Z71 SLE