Norotso

Or maybe you'll cook your transmission....

packplantpath

Yea don't do that. If it isn't shifting to a higher gear it is because it needs higher rpms to pull the load. The 5.0 pulling 4000+ rpms uphill in 4th isn't unusual. Let the computer control the transmission with tow haul turned on. And watch the transmission temps.

Buzam

Use tow/haul mode, that's what its there for. Maybe you underestimated the headwind, maybe you got some bad gas. 100km/h up a grade, into a headwind, towing 5500lbs is nothing to sneeze at.

I've tried out using Premium fuel when towing my trailer (similar size to yours) and it does make a noticeable difference in engine performance while towing. Might be worth a try.