I don't tow in the Rockies but I'm in an out of the Sierras all the time, having visited many of the most extreme grades. 30 minute slogs at 7+% grades, or steeper grades at elevation. The LX570 tows my 8000lb Airstream with authority. With 35 large tires and 15k+ full rig weight.
Yes, grades, elevation, and wind will all affect performance. Like any naturally aspirated engine, will lose some good oomph with altitude. Probably more than the F150 being boosted. But the 5.7L has deep reserves and will have no problem maintaining the speed and performance you demand of it (and are willing to feed it in gas). It won't sweat doing it, even in hot weather, while keeping 4-zone comfort with cooled seats.
One great thing you might see is that it'll engine brake much better on grades than the F150 as that's a common knock of the smaller displacement Ecoboost. The LX will have better stability than either GX or F150 being full time 4WD, prodigious curb weight, and strong bones. The AHC suspension can be an asset for towing as it maintains the perfect ride, constant height regardless of load, adapting damping and spring rate for laden or empty nicely. All the way up to 2000lbs payload if you need. Variable height to hitch up or leave hitched at site while being able to level fore/aft.
