Sorry, I got called to dinner while running the numbers,
OK, good news I think.
The max gross weight (the only way to deal with a camper you can't weigh) for the 233s Roo is 5,964 pounds.
At a 10% tongue load the weight carried by the Burban will be about 600 pounds.
At 13% (Optimum) you are looking at 900 pounds
At max tongue load of 15% you are at 900 pounds and that would be too heavy. So loading will be an issue.
The 2004 5.3L 1500 2WD with a 4.10 rear has a max gross weight of 7,000 pounds as you found out.
What concerns me is the Curb weight of your model Suburan is 5,323 pounds. That is you (at 150 pounds) and a full tank of gas.
You weighed the Suburban with your family and your "stuff" at 6200 pounds. Than means your family, your weight that exceeds 150 pounds, and your stuff weighs 877 pounds. Lets leave that number alone for a minute.
Assuming you buy a 10,000 pound rated WD hitch, it should weigh in at about 75 pounds. So say your rig with hitch weighs 6275.
That is 725 pounds available for tongue load. That is a load out of 12.2% and well in the safe zone.
Obviously you can't move your family to the camper, but for every pound of "stuff" you can move to the camper (careful not to exceed the GVWR of the camper) you can only improve your overall rig balance.
Remember a WD hitch does not CHANGE the tongue weight just moves it around between the front and trailer axles. Be very aware of your front and rear Gross Axle rating, but it looks like you will be "safe" towing your camper even at ITS GVWR. You just need to watch what gets put in the Suburban.
Hope that helps.