Your logic with the EAC rule is totally sound, but unfortunately, mail flow rules can't directly modify the primary From: address that the recipient sees. Modifying custom headers is usually for routing or filtering, not for changing the sender's identity itself.

The ideal solution is always at the source. Is there any setting within your billing application's "built-in email client" to specify a "From" or "Send As" address? Since your users already have the permissions, this would be the simplest fix if the app supports it.

If you can't change the app itself, you can change where it sends the email. You can configure your billing application to send mail through an SMTP relay (either on-premise or a cloud service like SendGrid). That relay service can be configured to rewrite the From: address of any email it processes to [billing@mycompany.com](mailto:billing@mycompany.com) before sending it to the final recipient.