Your 5.3 has 315 HP and 335 ft lbs of torque along with a 3.42 tear end. My 2013 Tundra had 381 HP and 400 ft lbs torque and a 4.30 rear end. Both yours and mine had towing capacity of 9800 lbs. The TT I had was about 7000 lbs on the road and 30' as compared to the possibly 8500 lbs or so (you will most likely be over weight) and 29' that you will be faced with.
A good chunk of my travels were in the mountains of BC, Alberta, Washington, Idaho, Montana, Oregon, and California. Power for the hills was never an issue, but in certain conditions, I was aware that my truck was just a half ton. Cross winds and passing semis moved me around because the truck was relatively light, and I had to anticipate what was going to happen and be aware that I would need to make corrections.
I then bought a heavier truck that pulled that same trailer with authority, and wind or semis were not an issue. Instead of "driving" the truck and correcting constantly in those conditions, the trip became so much more relaxed and at the end of the day I didn't feel like I had been working all day.
Up to you, I guess, but if I were you, I would be thinking about a ¾ ton like NanMedevac 70 has with more weight and more power. More weight to better handle the trailer movements, and more power to better handle the hills.
Safe travels whatever you decide.
2014 Keystone Laredo 292RL
2013 Palomino Maverick 2902
2018 GMC 3500HD, 4x4, 6.5' box, SRW, Denali, Duramax, Andersen
DeeBee, JayBee, and Jed the Black Lab
The hurrier I go the behinder I get. (Lewis Carroll)