Integrating a password manager with a browser is too fragile and risky way of using both. It is best to have them fully separated so they can't communicate. They should communicate exclusively via the user.
The login process should have some friction and should not be fully automated. Adding a secret domain-specific suffix to the password is very little friction for the user a gives obvious benefits: password manager does not know the password, it can't send it to other application (intentionally or by chance), it won't login the user by accident.