Tom,
Welcome and congrats on buying a tt!!!
Some mentioned the yellow payload sticker in the drivers door jamb of the 'Burb, post that info when you get a chance. This amount, "All passengers/ occupants and cargo must not exceed XXXXlbs" is what you should stay under for anything in the 'Burb: as it states, all passengers: how much does the family weigh?, cargo : iPad, iPods, coolers, firewood, tools, any added accessories after the truck left the factory, the wdh (weight distribution hitch), AND the tw (tongue weight)!. Depending on family weight, I would guess you will be ok with a 2500 'Burb.
The one limitation you have with the 'Burb is the max wdh weight of the receiver hitch! It is ONLY 1000lbs as you posted, due to being part of the crumple zone of the rear bumper. The "brochure dry hitch weight" is a fantasy number. From everything I have found, the 937lb tw you listed does not include the propane tanks or battery on the tongue! For that trailer I would guess it is 2-30lb tanks, if so, plus the battery add an additional ~160lbs to the tw before you even load the trailer. And the majority of the weight that is loaded in the front pass through (and under the queen bed) will go in the tongue as well! Our trailer has a "brochure dry hitch (tw)" of 920lbs (right around there), almost the same as your trailer's "dry hitch", and the same floorplan. Loaded we have about 1400lbs tw! And that was with an empty fresh water tank and empty black/ grey tanks. We are probably a little heavy in the pass through compared to some, but the weight adds up very quickly! And you will be about 1100lbs before loading the trailer with anything.
For a safe and stable tow, you generally need/want 12-15% tw, based on the loaded trailer weight. A lower tw (than the 12-15%) can and usually causes the trailer to sway.
All the other specs you probably will be fine, unless the family weight is on the high side or you want to take a cord of firewood with you. Just the darn hitch is the limiting factor.
The Reese HP Dual Cam, Reese Sc (hitch head has a little more adjustment than the Equal-I-Zer, very similar design), and the Equal-I-Zer 4 Way systems are very good wd hitches with integrated sway control.
Hopefully APT will post up his knowledge of the 'Burb since he has one himself. There has been a link for info posted where someone had the hitch reinforced to carry a higher tw. But that is removing the crumple zone.
Good luck!