I've towed a 6500-lb travel trailer through the Rockies with a 5.3 half ton Suburban, as high as 9,666 feet (Powder River Pass in WY).
Your axle ratio is going to make a big difference. Mine had the 4.10 axles. 3.73 should be acceptable, 3.42 will likely be problematic, even for a relatively light trailer.
I used tow/haul, and manually shifted gears. Be patient, both going up and down hills. Let the engine do the work downhill - not your brakes. Plan ahead and anticipate as you approach steep grades and sharp turns.
Regards to the controller - I've had great results with a Tekonsha Prodigy.