Hi, from NSW, Australia, so no local Irish knowledge i'm sorry. I guess you are looking for Irish research? If so, with respect, I suggest you mention what have you done already to save people repeating what you have already done.
For instance, Templemore Baptism Register seems to have survived for the Catholic Church http://www.rootsireland.ie/tipperary-north/tipperary-north-roman-catholic-parishes/ Searches can be made on Find My Past [free registration allows search & will show if a person by right name was baptised, & the year, & Diocese & Parish if I recall correctly]. My local Family History Soc have FMP subscription, so I could look up the ones you interested in. The Catholic Registers are online https://registers.nli.ie/parishes/0258 (if I have the right Parish; there is a map there), so probably well worth looking through around 1845. If you have any doubts about Templemore being correct, in NSW the Birth Certificates of Mary's children had to state where mother born [often just Tipperary, Ireland, but I find looking at every Birth Certificate gives a slightly different answer each one.
What else do you know about Mary's family in Ireland? Do you know Mary's parents' names? Married in 1865, so I presume not shown on the government marriage certificate, but what about the Church Register? That's about the time they started asking for parents' names I think. Not shown on Death Certificate? And what do shipping records show? Married at age 20 years, knew husband at least a year before that, so I guess arrived in Sydney or Melbourne around 1862-4. Needed a chaperone on board if not with her parents. So sister or brother on board? Gold rush period still? So nominated to be brought out by a relative already in Australia? In NSW, there are Deposit Journals which show who brought out. I'm presuming Mary didn't have the money to pay her own way unaided by a State Government.
Just my thoughts; I don't want to upset anyone, but I can't see how long ago you posted the above, so I thought I'd throw my 2 pence worth in.
Ron
