Select different sender email address without it being an official support email address

Hi All,

One of our clients has recently implemented encrypted email addresses for their customers on their own portal, so that we can finally email the customers directly without the workaround of having the client's support team forwarding our messages. However, we ran into the following issue and are seeking a solution for it:

The client's system doesn't accept messages from our registered email address because we also send system mails from that address. Therefore, emails from that address directly to customers are blocked. 

We're already using the Zendesk App 'Select Email Sender Address' but it doesn't solve the issue at hand: 

We need to be able to use a sender email address that isn't registered as an official one.

Is there any way we can achieve this?

Thanks!