Silverado Towing Capacity

I own a 2001 Chevrolet Silverado with a 5.3 V8 and 4.10 gearing. According to my owners manual, and everything else I can find, my truck is capable of towing 9,000lbs. My question is will I be doing any damage to the vehicle if I pull a trailer that weighs about 8,200lbs. We are in the process of purchasing a new camper for a trip out west. Does anyone have any experience or knowledge about this. The truck is in great shape, I maintain it well, it has 82K miles and has always been garaged. There is also a transmission cooler that was part of the factory installation package. I am the original owner. I am wondering if I will be doing any real harm or what to expect. I presently tow a camper that weighs about 3,000lbs. Thank you.