We towed our C19 with a 2003 Trailblazer for 4 years.
First your trailer is going to be way heavier than you think it will be. The brochure weight on our 2005 C19 is 2,851 pounds. However the brochure weight does not include any options on the trailer, things like the battery, propane, awning, or ac unit are not included. There should be a sticker that shows the actual weight of the trailer when it left the factory somewhere on the trailer. Our C19 weighs 4,400 pounds ready to camp, without water.
Now onto your tow vehicle. Power is not a problem. The short wheel base, P rated tires, and squishy shocks are going to allow the trailer to push you around whenever someone passes you. I upgraded to Bilstein heavy duty shocks. I got the highest load capacity P rated tires I could find, nobody made LT tires in the 16” size that came on it. A good hitch helps, but nothing is going to change the fact that your wheelbase on the tow vehicle is too short for your trailer. You cannot tow with the transmission on the overdrive position, the overdrive clutch in the 4L60E transmission will not handle the load. You are going to need an auxiliary transmission cooler to keep the transmission temps under control climbing hills in 2nd gear. Have the fluid and filter in the transmission changed regularly, I had it done every spring while we were towing with the Trailblazer. And last but not least nobody makes good towing mirrors for your vehicle; we used the universal strap on mirrors.
I have seen lots of people towing trailers with midsize SUV’s. It can be done, but be prepared to upgrade your Trailblazer to be better suited to the task. I would never go back to towing a trailer with the Trailblazer.
Gary