I'd suggest talking to a lawyer.

My (unsolicited) layman's take:

Someone from youtube != Google's legal department, which makes this situation both awesome and sad. If Google wishes to maintain their ownership of the youtube trademark, legally they must enforce their ownership of it (which means either granting you a license, or sending you a C&D). That said, I'd search around for trademark licensing info - they may have an existing license which covers this use.

In the mean time you may be able to protect yourself a little by adding some text that says that YouTube is a legally registered trademark of Google, Inc., and this site is not affiliated with Google in any way.