What do you think about this set up?

I had better luck than Curly had with my Tahoe but OP's set up is not ideal. You'll be at 7,500 pounds wet and loaded minimum. It won't be fun.

Quite a few years ago I had a very similar set up with a Chevy Tahoe, 350 CI engine and 3:42 gears. I did tow a 19' Wilderness and did two things. Ran out of power in the hills and burned up the rear diff. You will find out it is driving you and you are not driving it at the end of the day. The gearing is not right for the load you want to tow in my opinion.
Curly

It'll work. You absolutely won't be the first to the top of the passes.

I'd do it if the truck is well kept and not a pile of miles on it.
Just realize the limitations and drive accordingly.
I'd also want some helper springs/Timbrens/bags and min D load rated tires, preferably E load.

Dry weight is somewhat meaningless, particularly if you are stating brochure weights. I know my TT was much heavier than stated from factory, not a biggie since we had more than enough towing prowess in our 3/4 ton Chevy HD truck. I am sure your truck can pull your intended new TT, maybe even the space shuttle like a Toyota truck. It is not always weight but TT length when it comes to half ton trucks, flatlands may not pose as many difficulties, mountain and high elevation travel will certainly tax your engine and tranny. It all amounts to towing comfort for us, we live and do most of our RVing at high elevations, we enjoy having the power to merge and stay at posted speed limits, and the added safety of strong and capable braking system. We have friends and family with similar setups (half ton and 30' TT) that will tell you in a minute they wish they had more truck, it is ultimately your call, you could always upgrade to a more capable truck in the future. Our truck is not a daily driver and just sits idle until it is called to towing duty.

You haven't provided enough info to know if the truck is overloaded but...

In terms of pulling power, probably not a big issue on flat ground but expect to be turning a lot of revs on hills in a lower gear. As long as you have a low enough gear, it should make it to the top of the hill.

In 1958 my grandparents towed a travel trailer from NY to Texas to Denver and back. If their 210 hp Buick accomplished this you should have no problem.

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.

I towed a 28 foot 5000lb TT in mountains of northern California and Oregon with Ram 2500 V10 4.10 RED with great ease however I would not attempt towing same trailer in those mountains with my other Ram 1500 5.9 V8 with 3.42(?) RED.

I did tow this same trailer in flatlands of Texas with my other 2 1500 V8 rams with great ease. All three were older trucks. This may or maynot be a good comparison for you.