whats the truth?

Before we switched to a Dodge Diesel, we had the same truck as you bought. That is WHY we bought a diesel, as you will be lucky to tow 6000 lb. with the 3.42 axle. We were okay on flat and level, but here in Colorado we don't have much of that...that we want to got camp in. We had difficulty just going up long straight climbs, and certainly had trouble on big passes such as Monarch or Wolf Creek. We had to pull over several times to let the tranny cool, as well as the radiator...even with the HD tow package. Towing was always an ordeal, and we were exhausted after the strain of driving it, plus the right leg was tired of sticking into the floorboard. If you stay at lower elevations and flat, you can take it to the lake and back every weekend.

The max I personally would tow if I owned that truck would be a dual 3500 lb axle rig which probably has a gvwr of in the 7600 range.

However I would never buy that truck with the intention of towing that trailer, that is only if I had the truck first and that would be what I would personally set the limit at on the trailer. If I had a 7600 lb trailer and was shopping for a truck I would get a gas 3/4 ton.

But there are various ratings that you are already aware of that may or may not be exceeded in towing a 7600 lb trailer.

It's it a crapshoot as you say. You have all the numbers that you shouldn't go over. Your TV's GVW is the same as my empty SUV.

I'd say about a 22 foot light TT. That might be pushing it....

Sadly what you have is a rather large grocery getter. What you need to do is find the GCWR figures from the GM web site, Then load everything in it including passengers you would have for a trip and go to the scales. Subtract your scaled weight from the GCWR. That is the maximum weight trailer you can comfortably tow.
Don't even think about a fifth wheel, get a TT instead.