Authenticator refuses to use cloud backup.

I have managed to add the Microsoft's own authenticator codes to new phone, both can verify my Microsoft accoutn, issue is with the rest of authenticator codes stored inside the Authenticator, which I use for our numerous client environment logins.

Authenticator is unable to enable cloud sync, when using work email. Obviously our organization would not approve using personal emails for something this crucial for security. But for whatever reason the cloud login fails to see my work email as valid email.

I also tried to export the authenticator data into CSV, but this also gives "Export Failed: We couldn't export your passwords. Please try again later." with some "Sign in to Microsoft" screen behind this, this screen also refuses my work email. Giving "That Microsoft account doesn't exist." I guess "Microsoft account" here is intentionally confusing term, because, as far as I know my M365 account is also Microsoft account.

Can you provide me step by step instructions how to export or sync existing authenticator codes from old phone to new phone, without having to use personal Microsoft account (which should not be linked to anything work related). I will accept bulk transfer options, but grudgingly also one by one transfer, as long as I don't need to redo all accounts from their original sources spending potentially hours rummaging where the authenticator codes are regenerated.