For one particular website with implemented TOTP authentication, I have specified the secret in the card of that site in BW, by making the picture of the barcode in BW app on the phone - which populated the field with the otpauth:// URL .
Did the website prompt you to verify the TOTP code generated by Bitwarden? If so, where did you find the code that Bitwarden generated to submit for verification?
The website prompts for the code, which as I understand, is supposed to be generated by Bitwarden. The problem is that Bitwarden does not offer to generate it.
Have you tried one of the methods for enabling the TOTP authentication service in Bitwarden yet? Once it is setup, you should be able to generate codes.