When creating a Custom Connector that requires an on-premises data gateway, there are only 3 authentication options: none, basic, or windows.
If you try to sneak-in an API Key authentication by embedding a required-internal header in your Action definition, you get this error:
Would be nice if API Key was a supported auth type, or at least if the validation was altered to allow the Authorization header on gateway connections.
We are waiting for this feature as well. We really need to add our own "API Key" feature.