To do this, you need to set up Gmail to send emails via an "external" mail server. However, since the alias domain is already registered with Google, you can in fact set this "external" server to their servers and have it work just fine!
- Google Workspace Admin -> Apps -> Google Workspace -> Gmail -> End user access -> Enable
Allow per-user outbound gateways
in Domain admin. - Gmail -> Settings -> Accounts -> Edit Info (on alias account) -> Next Step -> Send through example.com SMTP servers (recommended for professional domains)...
- SMTP Sever: smtp.gmail.com, Port 465
- Username: [email protected] (Note: not the alias domain login, but your main Gmail login)
- Password: (Password for the above account)
- Select Always use a secure connections (SSL) when sending mail.
If you don't have this other domain added as an alias and are just sending from an additional email address you'll have to send via the email servers of this other domain.
(Yes, I know I answered my own question, but it took me an age to figure this out, so I wanted to share it!)