The easiest way to do this is to work with your Exchange admins to set up a server-side delivery filter that triggers on a special header and source. For example, a filter that says For all mail origniating from <your SMTP server> and containing the header 'X-AddToSent: yes', move to <recipient's Sent Mail folder>

Ideally, you'd implement some kind of strong authentication as well, so that no malicious individual can add to Sent folders by spoofing IP addresses. For example, you might cause your app to cryptographically sign the message and have Exchange use that as the criteria for determining the origin of messages.

Then, all you'd have to do is either (a)add an Fcc: or Bcc: header to the outgoing mail containing the sender's address, or (b)send another copy of the mail to the sender.

Of course, you could always just Bcc: the sender a copy of the message and provide instructions for users to set up filters instead...