Updated: 12.Feb.2024
MV3 doesn't support remote hosted code anymore so everything should be bind inside extension package.
Save Firebase Javascript modules inside /js/firebase/ of extension root.
E.g. files: https://www.gstatic.com/firebasejs/10.8.0/firebase-app.js
https://www.gstatic.com/firebasejs/10.8.0/firebase-firestore.js
E.g. for this case firebase-firestore.js at end of line #1 :
from 'https://www.gstatic.com/firebasejs/10.8.0/firebase-app.js'
to
from './firebase-app.js';
Then add firebase_config.js from below to the same folder /js/firebase.
Add inject.js inside root of package.
"Modify all firebase JS files to ensure all imports are from extension package, as remote script inserting is not allowed anymore in Manifest V3." In your case, it is ALLOWED. It is NOT allowed in the service worker. Inject inserts a tag script and it works quite well with remote scripts and without local ones (firbase-app, etc).