Darn. My first thought was to market a bag to go over the mirror and make millions. “The Pecker Protector!”. But yeah - a sock is a good idea. (Heh - cross-country ski racing wearing the thin suits we would had put a sock in our undewear when it got real cold).
Everyone has a sock lying around...or cut up a black plastic bag, ad a twistie...or threaad a string through it and make a pretty bow.