I think Supabase is fine. At least you are writing security logic in Postgres instead of Cloud Firestore Authorization Language. Last time I tried the Cloud Firestore (2019) security language, it was by far the most complex part of the app, and the local simulator behavior diverged quite a bit. I expect the Postgres bits of Supabase will be reproducible locally without the Supabase bits.
It's impossible to give a proper recommendation without additional details but Firebase will do just fine. There are tons of YouTube tutorials and Medium posts about how to do that.