You need to remember that the Catholic Church was an "illegal organization" until 1828.
Although there was still discrimination against Catholics in the first decades of the 19th century, Catholic churches existed legally under certain conditions.
The Catholic Emancipation Act 1829, which permitted Catholics to sit in the Westminster Parliament, and repealed the Test Act 1672 and most of the remaining penal laws, was the culmination of a series of Catholic Relief Acts beginning in 1778, followed by the 2nd Catholic Relief Act 1793 (Ireland). One measure which was not repealed by the 1829 Act was the prohibition on a Catholic priest celebrating a mixed marriage or a marriage of 2 Protestants. A priest in Ireland was arrested for that offence in the 1830's.
Marriages (Ireland) Act 1844 didn't apply to Catholic marriages.