I also believe it is "Carmel."
In a Catholic church one would expect any middle name to be derived from a saint and not be a family name. Carmel is a bit odd because it is a location and one associated with a female saint, but, nevertheless, I think it is "CarmeL"
The Carmelite Order has old and widespread ties to Ireland. No other short Catholic saint's name seems to begin "Ca" and end in "l." Also, it wasn't just nuns, but men in the order as well. Not to mention, "Carmel" was really the key word, with members of the order signing missives "Yours in Carmel."