Putting 6000-7000 LBS on top of a 1ton plus trailer (Empty weight: 2,210 lbs) requires a different platform. Maybe with a gas sucking 454... Otherwise you simply have the wrong tool for the job.
In your photo you can see the rear end squatted down and the front lifted up. This can change by moving the towed vehicle on the trailer as you change the tongue weight. The U-haul trailer really needs weight distribution bars over 5000 LBS by the hitch sticker. You care because the "balance" uses the rear axle as a lever and "lifts" the front tires off the ground. Reducing traction to literally lifting the front tires clean off the ground. When this happens the trailer likes to jackknife and take you off the road at speed - between power poles if you have any luck. Reduced traction for braking and steering also come into play. The harder you brake the more weight is thrown on the tongue and this lever action lifts the front wheels reducing their braking traction.
Even "IF" the vehicle can handle it you do not want to exceed any weights that the vehicle is "rated" for. It doesn't matter in non-CDL vehicles what the vehicle "can" do: the factory sticker and ratings matter. The insurance company and liability is yours if you are grossly overloaded. Nevermind the other driver was texting or other factors you simply are now fighting an uphill battle and your insurance company has walked away making atty fees out of your pocket...
Start with your GM dealer for tow ratings. If they come back ignorant you can get a good idea here
http://www.trailerlife.com/trailer-towing-guides/
FWIW a 1997 4x4 Tahoe with the 6.5 is rated at 7000 LBS.
Same year 4x4 2500 (3/4t) Suburban is rated at 7500 LBS.
454 engine is 10,000 LBS and meaningless. Can do doesn't allow you to do it...
K2500 Pickup 6.5TD is 8000 LBS. the 1 ton is rated less at 7500 LBS.
A higher tow rating is listed for pickups that tow fifth-wheel trailers. NA to the question, but, best I can tell it's 11,000 LBS with a 454. The 6.5 engine limits us for pickups, but, does better in SUV's.
You could find a HD truck and still have to get a better turbo for a 6.5TD to tow the weight. Myself I got a 2003 Dodge Cummins that can tow ~13,000 LBS aka 20,000 GCWR with the ~7000 LBS for the truck itself. And I still blew up the Cummins, but, at least it's rated to tow heavy.
The weight of the U-Haul car trailer is listed here:
https://www.uhaul.com/Trailers/Auto-Transport-Rental/AT/