This is the specs for the trailer you need to be concerned with:
Unit Dry Weight 5,847 lbs. (2652 kg)
GVWR 7,850 lbs. (3561 kg)
This is the Avalanche:
The 2WD 1500 offered rear self-leveling and a Z66 onroad suspension package, while 4x4 models could get a Z71 offroad suspension. Front side airbags and antilocking all-disc brakes were standard. GM's OnStar emergency/communications system was an option. An Avalanche 1500 could be equipped with "North Face" trim, created in cooperation with the outdoor clothing/equipment maker. Towing capacity is 8,300 pounds on the 1500 model, and 12,000 with the 2500.
You'll notice that if you load the trailer to the gvwr that you don't have much wiggle room.
I have a 1/2 ton Ford P/U and I use a prodigy contrioller and an Equalizer 4 point with sway control. The equalizer is extremely easy to use, and I have had no issues pulling, but my vehicle is rated at 10,000.
You could do it, but expect about 7 MPG and a struggle up hills. I highly recommend a tranny cooler if you don't already have one.