I recently towed a 5000 lb camper with a 2020 Tahoe with the 3.08 gears. I powered up long 7.5 degree grades without the transmission temperature exceeding 200 degrees Farenheit nor did the transmission downshift below 3rd gear (I have a 6 speed automatic) in maintaining 50-55 miles uphill without a stress. I have a built in engine oil cooler and auxiliary transmission cooler. On level ground with tow haul on the transmission held its gear and only on inclines did it downshift to 4th gear. Note, in tow haul mode the transmission is automatically locked out of overdrive. I got double d
igit gas mileage with the 3.08 axle and as high as 14.0 mpg towing 5000 lbs. My Tahoe is rated for 6400 pounds towing maximum. The point I'm trying to make is how easily the truck handled the load without excessive gear hunting and transmission temperatures averaged 190 degrees on a hot day. I have no problems recommending this set up as long as you don't go beyond your rated capacity. A 3.08 axle will work just fine of you stay within its limits. I normally tow at 65 mpg to maximize fuel savings but could easily tow at 70-75 mph of I wished. I don't see a huge advantage with 3.42 gears with my set up. Try a 3.08 axle and you will most likely be satisfied.

 

I tow a 7600lb Airstream with a 2018 Suburban Premier. It does fine on flat roads but on even the slightest incline, it struggles up hill. I wouldn’t attempt any mountains. You will quickly max out the payload and rear axle rating. I had the Suburban before buying the Airstream, but I will soon be upgrading to a 3/4 ton truck. You can opt for the 2021 2022 Suburban with the 3L diesel. That will give you the torque you need, but payload and rear axle rating will still be the issue.

 

Keep in mind those advertised trailer weights are not even close. Add in gear, batteries, water, options, etc. only true weight is scaling the rig with passengers and in truck load also. Trailer Mfgr. lie on weights all the time.
5.3 motor at the high end of load limit will struggle. 6.2 is the better choice. Emissions diesel and their issues are not my choice for long term ownership.
I don't know if they even make a 2500 class Sub anymore.
Your Airstream is a 3/4 truck tow vehicle for sure.

 

They do not make a 3/4 ton Suburban anymore. Agree 3/4 ton pick up is needed. 2500 HD will have ~3500lb payload

 

The Suburban already has a trans cooler, and you can monitor the trans fluid temp in the driver information center on the dash. You will need an aftermarket brake controller. But if you have not already purchased a tow vehicle, I would strongly consider a 3/4 ton truck as mentioned. You’ll be fine with the Suburban if you plan to take 2-3 trips a year and stay out of the mountains, but if you plan to use the TT more and get into the mountains, you’ll want/need the 3/4 truck. The Suburban has ~1500lb payload. Hitch weight can easily be 1000lbs. That leaves only 500lbs for passengers and other stuff in the Burb. Good luck!