Just did this my side (Arc10.4.1), loaded Nat geographic base map, zoomed to Ireland, then set the data frame coord sys to Irish national grid, then set either of these - TM65_To_WGS_1984, TM65_To_WGS_1984_2 as the transformation (TM65 is a Modified Airy based system). So, was just going in the opposite direction to your case.
No probs here.
So what version you on?