Towing with the torque converter unlocked for long periods (a minute maybe?) is not recommended. While an unlocked TC does provide more engine speed/torque than locked, it's a quick way to fail the transmission. FWIW, my 6-spd trans never locks the TC in first gear, and allows some slip unlocked in gears 2-6. Tow/haul mode in mine seems to hold the lower gear longer and lock up the TC quicker than unloaded/standard trans mode. I pay close attention to keeping it locked up.
Towing an RV up to 7k feet elevation is not easy for a tow vehicle. Get the trans cooler. Engine oil cooler and power steering coolers I would say are optional. As Tystevens said, the rest of the vehicle is capable.