It would be OK.  The longer the wheelbase, the better.

Ours friends had no problems, except for tires on the trailer, towing to
Alaska a 2 years ago.  8200 Miles.  Smaller pop-up, but they have now
replaced with a much larger one, and I have heard of no problems.

As always, it depends upon the year/age of the vehicle, what engine it
has (4.2L L6, 5.3L V8, or 6.0L V8 in 2008), 2WD vs 4WD, axle ratio, the
weight of the trailer plus supplies, where and how far you will be
towing, etc.

For 2008 models, the specs are at:

http://www.chevrolet.com/trailblazer/specifications/

under the 'trailering' tab.

The GCWR is the Cross Combined Weight Rating, which is the MAXIMUM
combined weight of the vehicle plus the trailer plus ANY passengers and
gear.  You will note that the maximum GCWR is 11,500 lbs, and the
maximum trailer weight is 6,800 lbs. See the NOTE which states that you
must use a weight-distributing hitch plus sway control for a trailer
over 4000 lbs.

Under the 'capacities' tab, you will note the curb weights of around
4,500 lbs, and the GVWR = Gross Vehicle Weight Rating = "weight of
vehicle, driver, passengers, optional equipment and cargo", of around
5,500 lbs (depends on model) meaning that the weight of the people plus
the gear in the vehicle cannot exceed about 1,000 lbs.

For the Maximum GCWR vehicle listed of 11,500 lbs, subtract the GVWR
Maximum vehicle listed of 6001 lbs, and you would arrive at 5,499 lbs.
This is the MAXIMUM weight of the loaded trailer you should tow, despite
the 'tow rating' of 6,800 pounds!!!  In other words, use the GCWR, curb
weight, and GVWR for your particular model to find the MAXIMUM weight
you should tow, and ignore the 'tow rating'!!!

Remember that knowledgeable people recommend towing only 75% of the
maximum weight as a safety margin, which would mean, in the case above,
4,125 pounds.  This would be the LOADED trailer.  The sticker on
trailers gives the curb weight, which at least sometimes does not
include the weight of common add-ons such as stove, fridge, A/C, awning,
full propane tank, battery, plus any and all gear you stow inside.  Gear
can easily approach 500 pounds!

Remember also to not believe the car dealer salespeople or the pop-up
dealer salespeople, as they commonly do not know, or lie, about what you
can safely tow.  IF you have done your homework as above, and then go to
a pop-up dealer and they give you the same advice, you have found a good
dealer.

Having said all the above, there are still plenty of decent pop-ups that
this vehicle can safely tow.

A weight distributing hitch makes a TON of difference...

john