The email client template should reference the table from which your users are sending emails. In this screenshot, it would seem that users are going to a record in "Email Client Canned Messages" and then clicking the send email action. Is that correct? Typically, the table would be something derived from the task table like incident or "time off requests" or something like that.
The way it works is that when the send email action is clicked on a record, the system will check for a matching client template that matches the record table and conditions. Because a quick message is applied by the user after all of the template calculations have been made, you wouldn't be able to do as your original request asked. If there was a way to use mail scripts within quick messages that would be a viable solution, but I'm pretty sure that mail script calls in quick messages are ignored.
The last option I can think of right now would be to have the quick message apply a watermark-type code to the message. Then you could have a business rule running on sys_email that finds inserted emails with type of "send-ready" and containing your watermark code. If it matches a record, it could alter the To and CC fields as desired.
