Run it as-is. I'm going to be towing my 9100 pound camper with my stock Chevy 350/5.7 and 700R4 transmission. With the 6.0 and the 4L80E you have, you shouldn't even know the trailer's there.
I honestly cannot say that I've ever hooked my camper up to the suburban. As of now, I'm just going off of specs to tell me what it can handle, and comparisons to similar vehicles that are rated to tow those kinds of weights. If it isn't capable of towing my camper.....I don't know what I'm going to do. But, I have total faith in that it's gonna drag the camper around with little issue.
Stupid question, but were you using a weight distributing hitch when you bought the camper? That'll make a huge difference in how the truck's stance sits and how it handles on the road.