Is that 1564?
Perhaps his mother was a Pack and his father a Fitche. All sorts of errors occur in transcription and then remain to haunt others searching.
I would think it pretty much impossible to make any connections from 1564 due to the amount of missing records in Ireland.
Dublin Will and Grant Books, Thomas Packe 1654 Diocese of Dublin 1270-1858
You might try the Registry of Deeds, Index, Ireland if they were people of means.www.rootsireland.ie
has Ballynakill, RC parish records from 1794-1899, marriages 1800-1898 and deaths from 1794-1815.
Griffith Valuation 1851: Queens/Laois/Leix www.askaboutireland.ie
Thomas Peake, Parish Kilmanan, Townland Castlecuffe, Poor Law Union Mountmellick
tenant of Rev. John Baldwin
#17b a house and garden 1 acres
There are Fitche, Fitch, Fitchie, Fitchee, Fitchey but only 12 in the Griffith Valuation, none near Laois.