Owned a 2004 Suburban 1500 with 5.3/3.73 4 speed for 9 years. It was an excellent family rig. It towed our 6,000 lb trailer just fine. I used a reese dual cam and always towed in 3rd gear. Climbing the mountain passes here in WA, I held the pedal to the floor and let it sing all the way up to the top. While we only towed a few times per year and within 500 miles round trip, it handled great. Traded it in last year on a 2013 2500 Suburban 6.0/3.73 6 speed. We will see how the new rig handles our new 8,000 lb trailer soon.
Good luck.