Last I knew, the Traverse was rated to tow 5000 lbs when properly equipped. Check with a service manager to verify this, and ask him what equipment was in the tow package. (As an alternative, if there is a Chevy Traverse owners' forum I bet they can tell you.) If you add the equipment that is critical, you can confidently tow the higher limit.
For example, my 2008 Toyota Highlander (very similar vehicle with 3.5L V6, 270 HP) was rated 2000 lbs without tow pkg and 5000 lbs with tow pkg. The main items in the tow package that enabled that rating boost were the extra transmission cooling capacity and a higher-output alternator (the latter is to keep up with the trailer's lights and charge line). I towed either a TT or a cargo trailer, up to 3500 lbs, at 60-62 mph for 140,000 miles with my Highlander and it still ran great when I sold it. The situation with your Traverse might be very similar.
I would recommend that the popup have brakes. If it doesn't, look behind the wheels for square backer plates with mounting holes; if you have those a brake kit can be added. Otherwise a new axle with brakes can be purchased for perhaps a grand.
Keep in mind that many members of this particular forum tend to think that you need at least a 3/4 ton diesel truck to tow anything. Your popup should tow pretty easily with its low frontal area. Get a class III receiver and whatever else the techies recommend, and go have some fun!