the GM 4L30E has a fluid temperature sensor somewhere in it, putting the transmission in "limp-home" if over-temperature. It will also change the torque-converter lock-up strategy as well, possibly at a lower temperature than required to go to full "limp-home"
Yep, you can see the fluid temperature in the live data from a Tech 2.
I believe it will lock up in 2nd on climbs when the fluid temperature is above a certain threshold whereas normally lockup will only happen in 3rd and 4th.
Haven't managed to provoke it either towing 1800kg of trailer tower up onto the downs nor on Stelvio with a bootful of camping gear. Not trying I guess.

General consensus is that the transmission cooling is adequate, and plenty of people on here tow.
However, it probably wouldn't hurt. If you fit the extra cooler before the heat exchanger in the rad. it will have greatest effect in reducing heat dumped into the engine's cooling system and also prevent the fluid being overcooled too far below the temperature of the engine coolant.
One essential point to look after these gearboxes is to given them a fluid change every 60k or so, though, as said.
Kevin