Another chance to learn the physics of automatic transmissions.
Most all vehicles today have a transmission oil cooler built into the bottom tank, or lower part, of the radiator, as standard equipment.
What make transmissions get hot, is slippage in the torque converter.
slippage increases when you drive up hill, or pull away from a stop sign.
As you gain speed, the transmission shifts into higher gears until you reach cruising speed (back off the throttle) and a lock up clutch in the torque converter engages so you have "direct drive" though to the wheels.
You can get along nicely with no added oil cooler, as long as you don't pull long hills in "Drive." Downshift manually and slow down on uphill grades to avoid overheating the tranny. Watch your engine temperature gauge (I hope you have one) and you will do just fine.
We pull our 16 ft deluxe Scamp
(fully loaded weight
is about 2800 lb) with a 2013 Highlander, and before that a 2007 Highlander Hybrid, and three Honda Odysses.
Never a problem.
You can help yourself with gas mileage by traveling with empty tanks (fresh water, grey water and black water. And avoid carrying heavy stuff in the back of your tug, to avoid suspension sag and weight
transfer off the front drive wheels. Happy Camping.