It's too bad the TSA employees don't bother to read the web site excerpt posted by Kevin. Still, with a doctor's note explicitly mentioning the mouthwash, you should be able to bring it on board, although don't be surprised if they insist on opening the bottle and testing it. You will also note that the TSA rules say you can bring water on but I have never read or heard of that being honored either. You might consider putting a sealed bottle of mouthwash in your checked luggage as insurance if you absolutely have to have it.
Finally, bear in mind that the few people who run into the even fewer TSA employees who follow these guidelines usually don't post on internet boards nor their personal blogs, while those of us who are treated like criminals by TSA do.
Charm