Does your vehicle have the towing option package, do you have the Tow/Haul mode and if so do you use it?
What octane of fuel do you use, have you tried higher octane on these uphill trips?
Problem is keeping your vehicle in the power band of the motor. More engine HP, or taller gearing.
More importantly, IIRC, you bought your vehicle somewhere south and flat (IIRC), have you ever had the battery disconnected / engine comp disconnected, had the vehicle 're-learn' your higher-altitude driving routine? It will make some (limited) accommodation for your higher altitude driving routine.
I believe there is a programming option to change and soften your shiftpoints. And If I'm not mistaken computer hardware available to swap that firmware on the fly, as needed.
Lastly, you know the hard shift and screaming RPMs are coming, so just hit the hazard lights and ease off and make the gear change manually when more appropriate. Exert some manual control on what's going to happen anyway. Don't kill your motor or trans trying to maintain a higher speed. Pull over and let backed-up traffic pass you when you reach your state's limit (in CA I think it is 5 vehicles stacked behind you)
IMAO it's these high load high RPM steep grade climbs that kill the 4L60. Trans temp is the key concern. Amend or improve that if you haven't already. I'm getting crazy low trans temps with my cooler install (verified with a laser thermometer, I have got the right data selected for my Torque app trans temp gauge). But I'm rarely topping 4-5k', not your 11k. 11k's rough on any vehicle not tuned for it, even computer-controlled and fuel injected. Even without towing a load.
So to answer your questions in order:
1. Yes, factory tow package with cooler and 145a alternator (alternator replaced in 2017.)
2. Octane: Yes, I'm a cheap SOB and typically run 85 (identical to 87 in the lower parts of the country) so I've never tried higher octane. I'm getting close to the bottom of the tank so I may give that a try, if for no other reason than to eliminate it as a possible cause!
3. Vehicle's "home" was originally Wisconsin but has been in Colorado since at least 2012, and battery has been replaced more than once so the ECM has certainly had time to "learn" the altitude.
4. I've gotten into the practice when climbing steep hills to go ahead and downshift manually to 2nd. Really, if it would stay in 2nd I'd be fine with it, the problem is that even in 2nd it sometimes bogs down slow enough to drop down to 1st for a few seconds and then jumps back up to 2nd. That's what I'm trying to avoid. Gears 3 and 4 aren't even a dream when climbing.
WRT trans temp, I'm running a SGII and I also have a trans temp gauge on the dash (had the dash rebuilt by Circuit Board Medics and had them add the trans temp gauge from the 6.0 V8 models - apparently all GMT800 trucks are actually wired for the trans temp gauge, but GM only put them on the trucks that have the 6.0 and larger motors.) On the SG I am seeing normal trans temps (on flat or intermittent up/down hills) in the 160 - 180 range. When climbing a steep hill, the trans normally goes up to about 210-215 but has gotten as high as 219. When I start going downhill again, the trans temps quickly drop below 200.