Connecting to google APIs for use later in a cloud function

I am developing an app where the user should authorize access to ones Google Account. Then, I am planning to store the auth_token in firestore so that later I can run a cloud function to send them a newsletter with data from google calendar (and some enrichments, but not relevant here).

I set up the Google API from the console and try to establish a connection with this Custom Action:

// Automatic FlutterFlow imports
import '/backend/backend.dart';
import '/flutter_flow/flutter_flow_theme.dart';
import '/flutter_flow/flutter_flow_util.dart';
import '/custom_code/actions/index.dart'; // Imports other custom actions
import '/flutter_flow/custom_functions.dart'; // Imports custom functions
import 'package:flutter/material.dart';
// Begin custom action code
// DO NOT REMOVE OR MODIFY THE CODE ABOVE!

import 'package:google_sign_in/google_sign_in.dart';

Future<String?> signInWithGoogle() async {
  try {
    final GoogleSignIn googleSignIn = GoogleSignIn(
      clientId:
          '...',
      scopes: <String>['https://www.googleapis.com/auth/calendar.events'],
    );
    final GoogleSignInAccount? googleSignInAccount =
        await googleSignIn.signIn();
    final GoogleSignInAuthentication googleSignInAuthentication =
        await googleSignInAccount!.authentication;
    debugPrint(googleSignInAuthentication.accessToken);
    return googleSignInAuthentication.accessToken;
  } catch (error) {
    return "error";
  }
}

The authorization window correctly pops up but in the end I get this error:

The OAuth token was not passed to gapi.client, since the gapi.client library is not loaded in your page.