Exactly as above. You have to use your own (or your domain's) authenticating SMTP server (instead of Google's server), otherwise Outlook readers will see your From line as "sent from
user@yourdomain.com on behalf of
user@gmail.com", which you don't want.
I'm going through setting this up right now as a test platform for a client. You can also set Gmail to reply as the address the message was sent to, so you don't reply as your personal account by mistake (this is in account settings just under where you add additional send-as accounts).