There is place (use Google) to get the build sheet on your Avalanche from the VIN. They vary by build.
The max tow on my Suburban is 5400 lbs unless it has the additional transmission oil cooler, then it is 7900 lbs (for 2009), for a 4x4. (200 lbs more if not 4x4) The tow guide has the same specs for the 09 Avalanche.
You'll max out the GAWR before the towing capacity.
The Tahoe is 16 or so inches shorter and has more towing capacity, its lighter but has the same engine and frame. But its short wheelbase is a problem for a long trailer.
2009 Suburban LT 1500 - 4x4 - 3.42
2009 Flagstaff 23FBS Super Lite
Nights Camped in 2015 - 21
Nights Camped in 2014 - 29
Nights Camped in 2013 - 31