I have been banging my head with the same/similar use case
Ghost + Outbound Only using Mailgun
After making needed DNS entries and messing with the Mailgun and API keys until the newsletter settings worked I still was not getting the expected outcome...
From what I have seen: https://forum.ghost.org/t/ghost-3-1-cant-get-mailgun-member-signup-working-on-my-ghost-install/10533/16
"The Mailgun configuration under the Members>Email Settings area is only for bulk email newsletters sent to members when you publish posts with the “Send email” toggle on.
The config.production.json email configuration is for any transactional email that Ghost sends, eg. Staff invites, forgotten password, member signin/signup emails, etc. This doesn’t have to be configured to use Mailgun but it should be set up to use a proper transactional email service if you want to get high deliverability.
Another source: https://ghost.org/docs/concepts/config/
I was able to locate and replace the advised settings but came to another barrier - it requires the Ghost CLI to run "ghost restart"
/home/cloudron/ghost# ghost restart
You can't run commands as the 'root' user.
Switch to your regular user, or create a new user with regular account privileges and use this user to run 'ghost restart'.
For more information, see https://ghost.org/docs/install/ubuntu/#create-a-new-user-.
Figured I would ask if this is best practice for cloudron or if there is another method before diving into attempting to make another user