When setting up an identity to send from a Google Apps (now Google Suite) domain alias, you may find that the messages you send either still appear to be from your primary Google Suite account, or as sent 'via' it. To fix this, please follow this guide to make sure everything is set up correctly on Google's end:

http://www.wphostinggeeks.com/add-google-apps-domain-alias/

In Mailbird, the identity should be created with the alias address in the email field, set to send through Google's SMTP server, and the credentials set to your main Google Suite email and password.

Note that if after setting everything up in Mailbird you receive an authentication error when sending, make sure that "Allow less secure apps" is enabled in Google, as that's what Google calls it when you authenticate using username and password authentication, which is needed for this to work. More info here: https://support.google.com/accounts/answer/6010255