I have a similar TV and am wondering how it worked out for awoodwaring.
I have a 2014 Silverado 1500 (5.3L 4WD Dbl Cab) with the "regular" tow package 3.42 rear axle, separate transmission and oil coolers, and brake controller. It's max tow weight is 9600lbs.
The trailer I am looking at is a 30'BHS style (KZ Sportsman) 6200lbs empty and 8200lbs full. I'm going to use a BlueOx SwayPro weight distribution hitch.
The issue, as I see it, is that the GCWR for the truck is 15,000, with the truck itself weighing 5200lbs, so if you add the empty truck and trailer (6200+5200) 11400, you get 3600lbs to spare for the combined weight.
That's actually not much once you add in people, the hitch, gear, the truck's fuel, etc. So now I am re-thinking the trailer, may need to go a little smaller.