My trailer (Cougar 30RKS) has a rated dry weight of 6,762. Loaded without water, I hit it's max weight rating of 8,200. But my wife brings a lot of scrapbooking stuff camping
I'm pulling with a Max trailering package. So I have a tow rating of 10,800, and consider it COMPLETELY irrelevant.
You will run out of payload long before you run out of trailer capacity. In my case, I have the 7,600 GVW and my truck weighed in at 7,540 loaded for 2 weeks holidays. So I was very close to the limit.
Estimate the trailer weight based on the loaded (dry plus carrying capacity) weight. Figure tongue weight of 12.5% (middle of the range) and add 100 lbs for the hitch ( I have the same one and it's not light). Compare that to the trucks payload capacity on the yellow sticker on the door jam. Whatever is left (if anything) is what you have for whatever you weigh over 150, plus all passengers, gear etc. in the truck.
In short, I really doubt the Suburban will be able to do it.
If you get a 1500, make sure it is the NHT package (either 5.3 or 6.2). While I am at my limit, I am happier with this truck for all the non-towing driving I do and do not miss having a 2500. I had a 2001 2500 with a Duramax and it was a harsher ride and didn't suit my daily commute (~3.5 kms). But it was a better tow vehicle. Some hills I am now limited to 80 kmh where with the diesel I could do 110. Not sure if the 6.2 would make a lot of difference there, but as I bought an lower trim level truck, I didn't have the option.
I live in BC, Canada so I know all about hills etc.