I'm on my third Tundra - a 2005 4.7L, 2013 4.6L, and a 2020 5.7L (all double cabs). Have towed with all of them to different extents - but not in the western mountain states. So for what it's worth:
Make sure you have working trailer brakes and a decent brake controller wired up. Your weight estimates sound a little low for the trailer and cargo mentioned. Is the trailer double axle? (Seems like a 14ft trailer should be). Anyway, with working trailer brakes and properly loaded your Tundra should be good at least double the weight you've estimated - but you'll notice the grades.
I did a fair amount of towing similar to the loads you're looking at with my 2013 4.6L. Helped my sister with a few loads of stuff when she moved about 6 yrs ago and moved myself 3 yrs ago from central VA to northern NY. Used a single axle 6x10 v-nose enclosed cargo trailer with a max rated weight of 3500 lbs - and had it loaded to that or a little more on some of the trips. I put electric brakes on the trailer and really appreciated them when trailer was loaded heavy, and especially on some of the trips made during winter. (remember to adjust electric brake gain setting to match actual final weight of loaded trailer!!)
Ok - performance. My truck accelerated and handled pretty good with the 3000-3500 lbs of trailer. You can pretty much forget about overdrive (unless going downhill!) My truck spent most of it's time in 4th or 5th gear in moderate rolling terrain. Keeping speed to 65 or below helped to keep it in at least 5th. The transmission was pretty good with gear selection - didn't search between gears much (my 2005 Tundra was a little annoying with that). Definitely remember to switch to the "Tow/haul" mode every time you start the truck - it resets transmission shift points for better pulling and improves engine braking. I'd guess that on a few of the grades you'll hit in NM and CO and up you'll be looking at climbing in 3rd or 4th in the 40-55 mph range. If you're used to driving TX at 70-75mph you should plan on moving a little slower
Gas mileage will suck even on flat roads - largely from the wind resistance of the enclosed trailer. My 18-20 mpg unloaded highway mileage dropped to 10-12 mpg with the trailer.
If you have the towing package on your Tundra does that include the dual temperature gauges - one for coolant, one for transmission? Nice feature - keep an eye on both when climbing the long uphills. Your towing package should also include a transmission cooler so another plus. If you don't have a brake controller already it's an easy hookup - the Tundra tow package is pre-wired. There should be a plug connection inside the kick panel down by driver's left foot and you can buy controllers pre-wired with the proper connector.
My newer 2020 with the 5.7L is a different animal - I've towed up to 9000lbs with that so far with no issues - but not in the Rocky Mtns!
