Do they really want to hear their own message played back to them first and then hear the response? That sounds really painful to have to deal with, especially after several replies. Ouch.
I can see hearing the reply and then, optionally, hearing the original message(s) after that just in case you forgot what you sent them or something. This really baloons the storage space if folks are doing a lot of replying, but it more or less follows the forward paradigm except the addressing.
But short story is no, there's no way to make Unity include the original message on a reply (just like replying on an email wont include attachments with it). They have to forward it if they want the attachment included.
As with all such things, make sure you ask your account team to open a PERs for it.