Note: The main post above is the solution, so read that, but I’ll mark this as such.
Bitwarden recommends using Flatpak for its Desktop App (or Snap), as noted in their docs here, but this seems to have addressed my issues when using the distro version as well (bitwarden 2026.2.1-1.1 from cachyos-extra-v3 repository).
Anyway, thought this might be useful for others who were struggeling to get Biometrics working, as was the case for me for quite a long time: Bitwarden Biometrics Checkbox Greyed Out Issue