Firebase Authentication in Extension for E-Mail Login with Manifest V3

Question

I'm trying to authenticate users using Email Login from an extension. I followed the documentation (https://firebase.google.com/docs/auth/web/chrome-extension) on this topic and tried to import it in my background.js worker file.

import { getAuth, signInWithEmailAndPassword } from 'firebase/auth/web-extension';

This doesn't work as I get the following error message:

Uncaught TypeError: Failed to resolve module specifier "firebase/auth/web-extension". Relative references must start with either "/", "./", or "../".

Now I tried importing it using CDN but then I get a security exception with my content_security_policy because I think Manifest V3 is not supporting outside scripts.

What am I doing wrong here?