In order for the [Send email to case assignee] Business Process to be executed using a certain mailbox, you would need to set up a support mailbox by following this Academy Article:
But please note that the automatic email message would be sent successfully only if the Assignee would have Access Rights for this particular mailbox. On how to setup a Shared Mailbox you may read here:
Hi we have some issues the the system is using different emails to send assignee notificaiton , but the email used to send the assignee notification externally does not correspond to the email used to create the case (which is what is needed / expected)