Just set up a couple of gmail accounts for friends' kids and registered their own domains for them. You can easily add the new email address to gmail so it uses their own domain (eg firstname.lastname@example.org) instead of a generic gmail.com one (look under Gmail Settings > Accounts & Import > Send As)
However using the standard setup which specifies using Gmail's own SMTP server to send mail out, the message arrives in Outlook with a 'sent from email@example.com on behalf of firstname.lastname@example.org. Looks a bit messy and gives away the gmail username.
You can stop that sent on behalf of from appearing by changing the Gmail settings slightly. Go back into the Send As setting mentioned above then click Edit info for the own domain name entry (email@example.com).
Specify that you want to use the domain's own SMTP but delete the smtp.kidsname.com entry and change to smtp.gmail.com (yes yes, same server as before, but it doesn't add that sent on behalf of label sent like this). Use the firstname.lastname@example.org login and password, specify TLS and port 587.