Email Forwarding - Send on Behalf without a Mailbox

Hello. I have a customer who has a domain set up with mail forwarding only, no mailboxes. But, they want to be able to send emails on behalf of the forwarding email address. It seems without a mailbox with User ID and Password it's not going to work. If we create a mailbox with the mail forwarding still in place, they will continue to get the emails forwarded to their other email address but copies of those emails will accumulate in their forwarding Inbox/Mailbox which they don't want to manage. Is there a way to create the mailbox and mail forwarding without the mailbox keeping a copy of the forwarded emails? I hope this makes sense. :)