I am not sure about the 2009, but the 2008 2500HD Crew Cab 4x4 with a 6.0 engine and a 4:10 rear max tow capacity is 12,300 lbs.
If it is a Duramax Diesel 6.6 engine V8 Turbo, the tow capacity is 13,600 lbs with a 3:73 rear, a 4:10 rear was not available in 2008 according to my brochure. I have a 2008 1500 Vortec Max heavy duty 1/2 ton with a 6.0 engine and a 4:10 rear and my max tow rating is 10,400 lbs. My camper fully loaded is 7,910 lbs, I use a Reese WD&S control hitch and the truck tows GREAT. I don't think you will have any problem as long as the correct hitch is selected and It is installed correctly.