Swap out your gauge cluster for an HD cluster, and you'll now have a working trans temp gauge. Use that as a guide as to whether or not you need to install an aux cooler.
I pulled trailers as heavy as 7,000 lbs with my '01 Suburban 1500, and never saw any trans temp that caused me concern. And that includes towing in the Rockies.
And yes, avoid 4th when towing. 3rd is a 1:1 ratio, so it puts the least stress on the transmission. Took a little while to get comfortable with the engine at 3000 RPM while cruising, since the truck had 4.10 gears. But I logged over 15,000 miles towing like that, and the truck now has 235k miles on it with the original engine and transmission.
2008 Suburban 2500 3LT 3.73 4X4 "The Beast"
2013 Springdale 303BHS, 8620 lbs
2009 GMC Sierra 1500 Denali (backup TV, hot rod)
2016 Jeep JKU Sahara in Tank, 3.23 (hers)
2010 Jeep JKU Sahara in Mango Tango PC, 3.73 (his)