How can I set up a BCC option for automated messages?
You could set up a second scheduled reminder for the membership and have it go to the administrator only.  You could even customize the message to be more concise and provide information that he or she needs (like the contact's full name and a link to the contact record on the backend) but wouldn't naturally go in the email to the member.

Although this wouldn't be the precise email that the member receives, it'll be driven by the same process, and you can set the same timing for it.