Although it seems as if Matthew did come from Yorkshire you may be able to prove he didn’t come from Ireland by consulting the religious census of Ireland which was taken in 1766 (in theory there should be no Cockcrofts on it! ) this was carried out by the clergy to determine the religious affiliations of the population. It contains Head of household, religion & number of children it is not a comprehensive record of all the population as those not eligible for payment of tithes were excluded.
Which leads me to think there were tithe maps drawn up earlier than the ones in England, the religious census was destroyed by the fire in 1922 but partial transcripts were made and are held in the National Archives.
Later we have the Griffiths valuation ( the first full-scale valuation of property in Ireland.) so called because It was overseen by Richard Griffith and published between 1847 and 1864. (this is after the tithes in England)
From Genuki: Ireland
Tithe Applotment books, Griffith's Primary Valuation records, and later land valuation records are also valuable census substitutes, because they record the names of the owners or occupiers of the land. These are described in the Irish Times site under Land Records.
From 1750 – 1832 only men with money (over £10 per year) and property rights could vote (actual landowners) and they had to be over 21
if Matthew met the requirements he may be on the pollbooks and electoral rolls
see more here:
http://www.thefamilyhistorypartnership.com/hints03.htmlyou used to be able to find burial records online at family history online but this website was taken over by find my past
the trouble also with the national burial index is that people have to transcribe parish registers and then give them to the people creating the NBI so this is not complete and remember you have 2 transcriptions already what one person may see as something someone else may see as something else so you may locate them in the NBI but it is always best to check the original parish register.
TNA has records of tax dissenters i.e. people who didn’t pay their tax for whatever reason
I think this maybe in section E192 but will need to double check this
TNA have many records of tax and i find this is the best place to start because after the civil war everybody has had to pay tax (even though it was originally created to put money in the coffers because we lost so much during the war and it is still here today)
TNA are very good but the trouble is not al the records are online and not all are searchable by name, they are getting there slowly though!
Did you see the new probate records on Ancestry? I am not sure if they cover the years you need (don’t think they do) but very interesting indeed, I have already found members of my family on there.