If the customer is using an Exchange server, then you can use either of the last two ideas.
Personally I would set an account in the domain for the copier to use and then setup a mailbox for the copier as well. This way you can also check your Exchange logs for everything sent by that copier.
One the copier, all you need to do is enter in the the SMTP Client details along with the account deatils you setup on the domain. For example:
Domain Username - copier
Domain Password - copier123
Mailbox for Account -
copier@example.com
You would use these details on the copier, therefore making it an authenticated user of the network / exchange service.