Strictly speaking, this is the Firebase team's issue - but I believe they've said before they don't have many engineers aware of Chrome Extensions, so it might be worth one of the Extensions team reaching out to them.
Problem
Firebase's Cloud Firestore SDK uses XMLHttpRequest, which Service Workers do not allow.
Obviously with MV3 coming, that'll be a hard stop on using Firebase's SDK.
Workaround?
I haven't tried, but presumably it might be possible to use a Sandboxed Service Worker and message in & out of it.
Other Info
The Realtime Database continues to work just fine. Every other part of Firebase I checked seemed ok.
More Info
If it helps, I can provide a sample extension to replicate it.