My trailer is about 5100 lbs lightly loaded and with no water (sticker says 4400lbs 'dry'). So its similar to yours when loaded, including battery, hitch, and propane.
My 09 Suburban 1500 is the 5.3l 6 speed 3.43 4x4 and is rated at 7400 lbs with the extra transmission oil cooler (5400 lbs without). My previous 05 Suburban was 5.3l 4 speed 4x4 and 3.43 which is like yours I expect. Same tow rating. The 07 refresh added 6 speed and improved the brakes and steering and more.
On a nice road I get 12mpg (Canadian, which would be 9 or 10 mpg US). Wind load is the big issue with the trailer height. Hills will push your RPM over 3500 to get the torque, and that will drain the gas quickly. I was below 8 mpg CDN in hilly or windy areas (and never passed anything ...).
Use tow-haul mode and watch the gas mileage computer to see where you can optimize the speed/gears. Inflate your tires to 45 psi at least (mine were 30 psi I think). I have P tires and will definitely get LT tires when I replace them. P tires are too squishy (due to sidewalls) for towing.
I have had 5 people in the Suburban (all gear in TT) and not noticed much difference in towing from having just 2 of us. At over 11,000 lbs GCVW, 400 lbs of kids doesn't add a lot. Just be sure to weigh the rear axle at a scale when loaded (check the rest too). That's where you will hit the weight limit on a 1500.
Overall you should be OK, but get a good WDH/sway control hitch and don't try driving at 65 mph.
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