Having a look at the database, I would say that I just need to change the code to:
$first_name = get_user_meta( $user_id, 'billing_company', true );
if ( !$first_name ) {
$first_name = $user_object->get('user_login');
update_user_meta( $user_id, 'billing_company', $first_name );
}
But I'm going to wait to see what wp client support says...