@krizzo I just closed this request, as it’s already possible to accomplish this by enabling “Ask for biometrics on launch” in the browser extension (Settings → Account security → subsetting in “Unlock with biometrics”):
(Optional) Check Ask for biometrics on launch to use biometrics when the browser extension first opens, skipping the initial unlock screen.
I never tried this option because my interpretation of the wording “Ask for biometrics on launch” was different: I thougt the browser app is launched with the browser and not every time I unlock the vault. But it does exactly what I was looking for. I just would have named this option “Skip unlock options for biometrics” or similar. But now I know it, thank’s for your quick reply.