This is the sticking point, it is Google's causing, not App Inventor, that will not allow a user to authenticate in a webview (even if you do try to change the user-agent). The user would have to go out to Chrome, authenticate, then back into the app, and use a webviewer to "sign in", then send requests to a google apps script web app to perform calendar functions on their own calendar/s.
Although this is not for the google calendar, see here for how I approached this for Google Drive functions: