Aren’t you supposed to have your personal Bitwarden account and get work passwords shared to your account? I thought that’s how Bitwarden for organisations worked.
It doesn't take me many seconds to swap accounts. LastPass allows you to be signed into two accounts at the same time in the same browser?
I'm using that on my VaultWarden server to share data between different accounts and it works well for me. This may not work in your specific situation if your company manages your Bitwarden account, though.
Not sure why the web extension doesn't. Might have something to do with autofilling or adding credentials to HTTP Basic Auth?