We have a '10 Suburban 1500, so very similar to the OPs. Ours has the K5L package installed, 3.42 gears, etc. Our trailer is 6000-6500#, tongue weight around 800# when water tanks are full. We have a family of 6, but our kids are all small, so our total family weight is about 500#.
The combo does fine. We don't win any races. We move along in gears 3-5 most of the time, with the occasional dip into 2nd going up a decent hill. Not my ideal package, but it gets the job done. Most importantly, it has never felt unsafe or unstable in any respect.
One thing though -- I don't know that you can install a heavier hitch on it. The hitch on ours is basically built in to the cross member at the rear of the frame. Seems to be an 'it is what it is' situation.
For what it is worth, I can't imagine full-timing in a trailer that the OP identified, at least with a family of 5. Our TT doesn't have a slide, but is otherwise quite similar, and there is major claustrophobia when we are all inside that thing, even only for a couple days! I like the OP's ambition, but I would think you would need a significantly larger trailer to avoid going crazy. At least, I would.