Facing the same issue building an extension that uses Firestore.
I hadn't used npm or webpack before this, but since you can't use <script> tags to access the code from gstatic I had to learn both. But then I hit the issue that I was using v8 not modular v9 code in my app/extension and v8 doesn't play well with Chrome extensions because of the use of xhttp so I had to upgrade to v9. I tried the compat libraries but nothing was working, so ended up building the app back from scratch using modular v9.
I am now back to a working v9 Firebase app with Firestore using npm and webpack.
Next I am going to use chrome.identity to get a Google token, then pass that to Firestore to authenticate the user.
Once I get that working I will update you here.
But I agree, it seems way harder than it should have been to use v3 Manifest and Firebase together.