Is it possible to show a different WP menu when using password protected pages?

If you want to check if a post has a password, and that the password has been entered, check both post_password_required() and $post->post_password:

$post = get_queried_object();

if ( ! post_password_required( $post ) && ! empty( $post->post_password ) ) {
    // Post has a password, but has been entered.
}