Are you sure about this? In the documentation, it says that the “Bitwarden desktop app must be logged in and running”, but it doesn’t say that the desktop vault must also be unlocked.
If the desktop app must be unlocked, then you would have to do the biometric unlock twice (first on the desktop app, then on the browser extension), and (as @Peter_H suggested) ensure that both vaults are locked if you leave your computer unattended.