The Bitwarden Web app (a.k.a. “Web Vault”) does not have biometric (fingerprint) unlocking capabilities, so my best guess is that you are actually describing the Bitwarden browser extension. Please confirm.
When Bitwarden prompts for the verification code, does the phrase “We don’t recognize this device” appear anywhere in the prompt? Alternatively, do you see an option (button) labeled “Use your recovery code”?
When you “log into” your Bitwarden account on your phone, do you see an Unlock button or a Log out button anywhere on the authentication screen?
Finally, I would highly recommend that you use your phone to create a vault export using the format “.json” — not “.json (Encrypted)”. Store a copy of the export file in a secure location (e.g., on an external USB drive) and then delete the original file from your phone.