We owned a 2008 Chevy Trailblazer which is basically the same vehicle. I have no doubt the engine and tranny can do it, but I would suggest you read some of the "stickys" to get a better idea of your actual tow rating.
I would guess that trailer weighs closer to 4000lbs, you can't really trust dry weights. By the time you load it up you will easily hit 4500lbs or more, at that point your tongue weight will be between 450 and 675lbs(10-15%). If I remember correctly, the payload capacity of the TB was right around 1200lbs, so if you take 600lbs off for tongue weight you have at most 600lbs left. Depending on how many will be riding in the vehicle, it may work out, remember that 600lbs has to include everyone and everything in the vehicle including a full tank of fuel and this weight has to come off of your maximum tow rating.
We had considered pulling an X17Z with the TB, which is a smaller trailer than you are looking at, but with the four(sometimes five) of us and all the stuff we end up taking, it would have put us over the GVWR of the TB.
2013 Starcraft Travel Star 229TB
2011 Jayco X17Z(traded in)
2010 Jayco Jay Series 1207(traded for the 17Z)
2011 Chevrolet Silverado 2500HD Z71 CC 6.0L/3.73
Reese "Pro Series" 800lb Trunnion WDH w/sway control