I installed an external transmission cooler on my 2017 Yukon before setting out on a cross country trip last summer towing a 4K pound trailer. The dealer told me that the transmission cooling valve opens at 195F, and my towing experience on the flat confirms that (transmission temperature rises to 195F and then stays there going no higher). While towing through the Rockies from Canada to Mexico up and down long 6%/7%/8% grades the transmission temps sat at 195F most of the time, reaching 210F a couple of times for a few (10) mins while climbing mountains in the southwest and then dropped back to 195F on the descent.
The external cooler worked like a charm. Because the transmission was running cooler it wasn't dumping all that extra heat into the radiator which kept the engine coolant at normal operating temperature even during the climbs. I'm confident that you'll do fine with the added external cooler.