I research the Irish Family History Foundation's database. It contains 12,800,000 surviving rcds for abt 70% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800. Cos. Donegal and Derry were recently added to the database.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 30%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are ONLY available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome).
For Cat(herine) King M 1843 +/- 4 yrs somewhere in Ireland, I found 20 rcds with 3 in 1843 (exact). For these 3 the parishes are Louth, Limerick and Galway. None of the 20 are in Tyrone.
For Fr(ancis) McShane M ditto, I found 1 rcd in 1841, Co. Tyrone.
For Fr(ancis) McShane B 1815 +/- 4 yrs, somewhere in Ireland, with the father's forename unknown, I found 2 rcds: 1813 and 1817, both in Co. Armagh. The earliest B rcd for this name in Co. Tyrone is in 1828. It looks like these ancestor's rcds didn't survive except in your bible.
For Cat(herine) King B 1827 ditto, I found 43 rcds, with 3 in 1827 exact. These 3 are in Westmeath and Meath counties.
For Catherine and Francis' offspring, I found NO rcds matching even approximately the DOB you listed.
Conclusions 1) the offspring were likely all born in America. 2) The marriage of Catherine and Francis was also likely in America. 3) Rcds of the B events for Catherine and Francis either weren't rcd'd, or haven't survived.
Sometimes the LDS Family History Library and National Archives of Ireland microfilms contain B/M/D events which didn't survive to be transcribed into the IFHF database. Your chances of finding genealogical rcds for your family at either of these are slim.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.