I'm going suggest that it's not a great idea, but opinions vary and some people tow trailers like you have with one. I tow with a 2012 3.0 TDI, but at a much lighter 4,500 lbs fully loaded. The problem is not power, but tongue weight capacity. I don't see how you would even stay under the rated 770 lbs tongue weight with a 7,000 lb trailer (is that dry or loaded?). I'm way under that at about 550 lbs and I wouldn't want to go any higher than that. We use a WD hitch with 600 lb bars pretty much maxed out. This is a unibody vehicle with the bumper bolted to the unibody, so beefier WD bars are not something I would feel comfortable with, although opinions vary here.
Sway is not an issue. Although it is a short vehicle, it actually has a decent wheelbase at 114". That's only 2" less than a 2014 Chevy Tahoe.
Look at the offroad and towing section of the
Club Touareg forums for more opinions and info.
We love the Touareg, and it's perfect for our situation right now, but if I were you I would keep the suburban and use it just for towing, or get a truck.