I would suggest that when initiating a browsing session during which you don’t want any interaction with the Bitwarden browser extension, use the browser’s Extension Manager to temporarily disable the Bitwarden extension.
FYI, the prompting to save logins even when the extension is not unlocked was the subject of a popular feature request from 2018 to 2021, and the absence of this feature caused unnecessary strain on Bitwarden’s customer support and bug reporting system, as explained by @djsmith85 in his comment above.
Anybody who would like the current feature to be reverted can vote and comment in the following active feature request thread: