If there were multiple monastaries for Catholic factions, the Gaels shouldn't have monastaries to specific saints except Padraigh (Patrick). Otherwise, they had;

Monastary of the White Oak; Technically a monastary to Saint Patrick, it was a dual monastary that housed both monks and nuns, and the order was legendarily founded by Saint Bride, Patrick's favored student. It was found around a great white oak that was originally a pagan Irish holysite, to convert the nearby families, which was successful. The order did little specific; they produced doctors, missionaries, but most often intended to do general good works and assist however was needed.

Monastary of the Green Cross; The Order of the Green Cross was actually a vaguely military facility, founded by Saint Finbar in Connaght after his conversion by an unnamed priest. Saint Finbar was a soldier, and an aristocrat. The order was a kind of forerunner to later military brotherhoods, with looser rules and hierarchy; the aristocracy in the regions would sometimes prove their devotion by 'joining' the monastary, ostensibly, as a 'lay brother'. They would provide arms and armor, and were called 'Criosakinna'; 'Cross-Kin' or 'Christ-Kin'. They would also wear a veil of chain armor or scales, to hide their face, representing the surrender of themselves and their personal glory for the good of the church and the Christians in the region. Also was founded the concept of Christian 'Rastriagha', contemplative 'berserkers' who spent their lives in study and learned self-control and the ability to enter a trance-like state described as 'physical emptiness'. This building, in game, would be meant to train the Rastriagha. As an aside, the 'green cross' was eventually absorbed into the Templar, but existed longer. The association with green and Catholic-Irish stems originally from this order. In Scotland, the Criosakinna and Rastriagha never really caught on, but these monastaries did exist, and the monks inhabitting them would have been some manner of 'warrior monks', but don't know if that's good enough for a unit (considering they only appeared to have ever fought pagan Picts, and maybe some Norse).

Monastary of the Ascended Virgin; Similar to a saintly dedication, but to Mary. The order was founded at Paroghan hill, the site of a Marian vision. It was the first monastary in Ireland built in the 'traditional' sense. Earlier monastaries were like small villages founded around the hermitages of the green martyrs; hermitic monks who went into the wilderness. They would be a series of hovels. This monastary was a large square edifice mostly made of wood, reinforced with a thin stone outer wall. This monastary was essentially a type of hospital, and others of the order sprang up in other regions to assist with illness and the many wounded and infirmed spawned from wars with Norse invaders.