During development, the application/extension ID is persistent as long as you load the application/extension from the same path, if I remember correctly. So during development, you can just register a Google OAuth application for that ID.
Once you decide to put your application/extension in the Chrome Web Store, you publish it only to testers (or anyone with a link), install it and get the web store generated key using the instructions you followed.
Alternatively, you can pack your application using the "Pack extension" button and install it by dragging the generated file into chrome:extensions. This will generate a key.pem file that you can later re-use for future updates to your locally packed application.