Can I set an Reply-CC in the same way I can set a Reply-To

Emails come from the website with a FROM address set to our email domain to avoid spam.
We set the reply-to to the customers email address so when we hit reply, it goes to them (not us)
Is there a way to set a reply-cc to copy our support email so that when someone replies, we’re all automatically notified to avoid duplicate replies?
We try to do that but regularly someone will forget and we end up with duplicate replies.

Hi Shannon. If you send a copy of the notification to your support email address, and then you REPLY ALL in your email client, there will be a copy sent to all the email addresses who received the original email. Will that work for you?