It just seems strange to me that you'd have to get a dispensation for a 3rd cousin marriage. Third cousins are not very closely related, and it seems like in small communities in rural Ireland in the 1800s everyone would have been third cousins. Then again, I don't know anything about Catholicism nor rural Irish communities in the 1800s, so...
On the other hand, they would not be likely to obtain a dispensation for a 1st cousin marriage unless there was a grave reason for the marriage and without permission from the Vatican, which would have taken a long time and been expensive.
The priest also had to take into account any previous marriages and relationships between the families and set them out in the dispensation application. A parish priest could grant dispensations for more distant degrees of consanguinity; decisions on closer degrees had to be referred upwards and cost more.
I imagine working out all these relationships would have been part of the skill-set of a matchmaker.
Kilmovee is a rural parish in Mayo. Plenty of marriages involving 3rd and 4th degrees. A few 2x3 which I think is first cousin once removed.
https://registers.nli.ie/parishes/0137Marriages 1824-1848 microfilm 04224/02
Marriages 1854-1880 microfilm 04224/05
1840
page 54; 20th Jan. Thos ? Gordon & Briget Grady 2x3
p. 55; 7th Feb. Ant? Andrew? Grady & Mary Kirins 2x3
p. 55; 17th Feb. Jas Dalton & Mary Forkan 3x3
p. 55; 27th Feb. Pat Duffy & Winy Duffy 3x3
p. 57; 2nd. March John Glavey? & Honor Kine 2x3
1841
page 60; 22nd. Feb. Michael Duffy & Winy Duffy 2x3
1856
2nd. Feb. Patrick Duffy & Maria Duffy 2x3 consan. (Written after this was "paid bishop".)
February was most popular month for weddings.
Duffy was a common name. I don't know if the Winy Duffy who married Feb. 1840 was the same woman who married a year later or if the Pat Duffy of 1840 and 1856 were the same man.
None of my Irish ancestors whose marriages were in church registers seem to have married relatives so I have no personal evidence to offer.
I wish the priests who conducted the marriages of my English Catholic ancestors had recorded degrees of consanguinity to give me some clues as to how they were related. I have 2 lines with recurring surnames, including 3 pairs of brides and grooms with same surnames.