Surely the Irish version of a name has no more of a legal status than a nickname or any other AKA.
If your name is John Byrne, you have much right to call yourself Osama bin Laden as you do Sean O'Broinn (assuming I've spelled that right). John Byrne appears on your birth cert so why should you be asked to identify yourself as anything other than your real name?
If someone was to ask you for,say, the Chinese, French or Spanish version of your name, would you feel equally compelled to find out what it was?