It's not possible to update an un-editable field via Airtable API if the Airtable user doesn't have permission. One user at least has to be able to edit the records directly on Airtable. The Airtable API access is tied with the Airtable user account. So if you have an Airtable account in your base and you choose to give field permission only to this specific user, you'll be able to update that field value via miniExtensions or any 3rd party apps.