I’ve been running Bitwarden without problems on the macOS Ventura beta with Safari 16 and 16.1 for a few months now. It’s still working with yesterday’s final release.
Can you give a little more detail what is not working for you?
I had already run this command but it didn’t seem to help:
/System/Library/Frameworks/CoreServices.framework/Frameworks/LaunchServices.framework/Support/lsregister -kill -r -domain local -domain system -domain user
The step I missed was to navigate to Safari / Settings / Extensions, then to check the box to enable Bitwarden. (I was looking under Safari / Extensions, which just leads one back to the Apple Store app.)