It's quite possible that if Mulligan was Presbyterian, he was descended from a mix of Catholic and Presbyterian (probably originally Scottish) Irishmen. This was far more common in 18th and 19th century Ireland than you'd think, due to intermarriages and religous conversions. The politician and philosopher Edmund Burke, who was from Cork, the same county as the Mulligans, was Anglican/Episcopalian but his family were recent converts from Catholicism. The playwright Richard Brinsley Sheridan came from a family of Gaelic chiefs who converted to Anglicanism in the 17th century. These are just two very well-known 18th century examples, but my basic point is that tribal/religous identity in Ireland was more fluid than stereotypes or politics try to make us believe.
Also, many Irish Catholic immigrants in the 18th century converted to Protestantism in America in order to get ahead. Philadelphia was the only city in the colonies with a Catholic church, and many people were virulently anti-Catholic.
This may be more than anyone wanted to know about religion in 18th century Ireland and America, but oh well...