Which while not exact is awfully similar and can maybe help point in the right direction. The harp is the traditionally irish symbol, since this is clearly when the Republic was part of the UK, maybe yours is after when they became independent. But I'm not sure how military buttons work, so this is mostly speculation. I'll keep looking.
Note: I assume we're talking 'buttons' (clothing) rather than 'badges' (What UK/Irish folks call pin buttons...like the decorative kind)