Ireland can fall into the sea as far as I’m concerned. They and their nutty president hate my country. I have no use for them.
Joining the Commonwealth would not alter the constitutional status of the Republic of Ireland in any way, still less make its citizens 'subjects of the monarch in Buckingham Palace'. Plenty of other Commonwealth member states are also republics. The Commonwealth is a very loose grouping with no constitutional conformity, and only some members also choose to have the UK monarch as head of state.
....not that I'm suggesting that this is either remotely likely, still less desirable.
By the way, today is the 1000th anniversary of an important event in Irish history—the battle of Clontarf. Brian Boru, the namesake of the O’Briens, was killed in the battle.