Good idea. I also think that the advantages would outweigh the disadvantages. It would need to be 'optional' for sure.
However, from my basic understanding of public key cryptography I can't conceive a method to implement this without affecting anonymity. Moreover, as the 'service provider' you are already in contact with your customer. Therefore, why 'leak' some of this information with other bitcoin nodes.
It would be a bit like signing a 'mixminion' email with the actual 'from' address.