I see that Feargal has already been working on the marriage source and changed the link to FindMyPast, where seeing the image requires login but not a paid subscription.
I note that Patrick's death record and tombstone inscription give different ages at death. "17 March" was an extremely popular estimated birthday for Irish people who neither remembered nor knew nor cared exactly when they had been born, particularly for Irish men named Patrick, since it is St. Patrick's feast day and one of the few dates celebrated annually. For similar reasons, "4 July" also crops up extraordinarily often as an estimated birthday in the United States.
I don't see any source showing that Martha's last name at birth was Doyle. If there is no source, then it should probably be changed to "Unknown".
There is also the possibility that the grief-stricken relative registering Patrick's death misunderstood the question asked and gave Patrick's wife's name when asked for Patrick's mother's name, a surprisingly common error.
The two Marthas have become confused in user-donated family trees at ancestry.com, including what may be your own at
https://www.ancestry.com/family-tree/person/tree/193991288/person/142536340756
where the mother's estimated birth year is several years after her son's, citing her daughter-in-law's 1860 census return as a source.
I also recommend copying your DNA data from 23andMe to GEDmatch.com, MyHeritage.com and FamilyTreeDNA.com and adding the relevant details to your WikiTree profile. Then search your matches on all the DNA websites for those with ancestors named Doyle and see whether any of those ancestors might be a sibling of Patrick.
You could also persuade your father to swab for Y-DNA analysis by FamilyTreeDNA and see if any close male line Doyle relatives show up in his matches.
Note that Carnalway civil parish, where Eric found an Edward Doyle in the Tithes, is covered by Kilcullen Catholic parish at
https://registers.nli.ie/parishes/0483
There is a gap in the surviving/online baptismal records from 1 Sep 1818 to 25 Apr 1829, which may cover Patrick's birthdate.
Your ancestors appear to have emigrated shortly after their marriage during the Great Hunger of 1845-52, so you might want to check for Doyles in eviction and workhouse records from Naas if such have survived.