I think I've figured this out...
When you elect to use the Google App Signing process, Google Play CHANGES your uploaded app's signature to the SHA-1 certificate fingerprint of the App signing certificate (as against the Upload certificate). See Figure 1 at this Studio document for the process flow.
[Btw, you can find both app's certificates at Google Play Console -> Release Management -> App Signing]
This means that, for Oauth to work, you must create a new OAuth 2.0 credential (at console.developers.google.com) that uses the App signing certificate's fingerprint (as against the Upload certificate's fingerprint) with the same package name.
Your package will now have 2 Oauth credentials, one for production releases, the other for your development/testing. Both will have the same package name; the production credential will use the App signing fingerprint, while the dev/testing credential will use the Upload (or local keystore) fingerprint.
Obviously, you will not know the App signing fingerprint until your app is first deployed to Google play (GP assigns it). So, the production credential will need to be created after uploading to Google Play for the first time, but prior to clicking "Rollout to Production".
与恶龙缠斗过久,自身亦成为恶龙;凝视深渊过久,深渊将回以凝视…