It turns out that chrome.identity.getAuthToken was still working however when trying to use the app from the Chrome Web Store, there seemed to be some sort of cache that rendered the previous version of the app which cleared after completely removing the app and re-installing it.
To conclude, you need not (and can not) use the "key" value in the manifest.json for the deployment to the Chrome Web Store however you do need it to run locally so two versions of the manifest.json need to be used; one for each environment.