The MemberMouse system uses the wp_mail function and must be sent through your server's associated mail service.  

You can however Change the Email Address your MemberMouse Emails are Sent From.  

You can also select specific email addresses for: 

• Welcome Emails in MemberMouse > Product Settings > Membership levels > Edit (pencil icon in the Actions column) 
• Push Notification Emails in Developer Tools > Push Notifications > Edit push notification (pencil icon)