I don’t know if privacy-centered apps prevent access to Biometrics data, but I tried other things that seem to work nice in Thorium with fingerprint.
The browser does not actually see any biometrics data or invoke any biometrics APIs. It sends a request to the desktop app and the desktop app handles that.
The issue why it is not working is because the Bitwaden desktop app needs to copy a file manifest file into the browser directory for inter-process communication to work, and these are hardcoded in the Bitwarden desktop app. This could be changed in the Bitwarden app source code.
Alternatively, you could probably copy the manifest from a working browser (chrome) to a non working browser’s directory. This should also work, though I haven not tested this.