6200 pounds is with no accessories added to the vehicle, no cargo, no passengers. Just a 150 pound driver and a bit of gas. So, as long as the weight in the truck is less than 500 pounds, you may be able to tow 5700 pounds. But if it is empty, why bother towing it; an empty RV is not of much use.
You sure you will never have more than 400 pounds of food, water, sewage, clothing, tools, hook up supplies, utensils, manuals, tapes/dvds, games and gadgets?
Hey baxter08, You won't have trouble towing the trailer on a flat road. You can probably even get up the hills, SLOWLY. You may increase long term problems, with your brakes, drive lines, differentials, transmission, and suspension. It would be best not to tow at maximum load capacity. If you do it occasionally, for short distances you can get by. Just remember what you get moving must at sometime come to a stop. Don't drive too fast, your brakes are designed to stop your Trailblazer and a 6200 lb towed load, and they won't stop it on a dime.
With 7100 pounds capacity, you may be ok, as long as the amount of stuff in the trailer and the stuff in the truck (including people) is less than 1400 pounds. Sure you can go over the limit, but it increases your chance of an accident, breakdown or excessive wear of your truck, and increases the stress of your trip. Not to mention that if you do have an accident, the lawyers may be all over you.