Looking at the Lucas directory of Cork 1787 I see that two Fittons are in business as mathematical instrument makers in Paul Street Cork. Francis is at number 36 and Alexander is at number 60. As Fitton is not a very common name in Cork it may be worth following up on these. A number of old Cork trade and postal directories are free online at www.corkpastandpresent.ie/places/streetandtradedirectories/
I also note that a number of entries are made in Sir William Betham’s abstracts for William Fitton Cork- these can be consulted at the National Library of Ireland in Dublin, they have also be filmed by the Mormons and have recently been added online to the Irish records on FindMyPast.
Thank you for your suggestions, JillWill. Betham’s note no.157 confirms the names of William’s five sons that my grandfather’s cousin provided in a pedigree in 1903, namely: Francis, Nicholas, Alexander, George, and Walter – the first and third being the sons listed in the Cork Directory of 1787 – but in
addition, Betham listed a sixth son William, and a daughter Mary – both being new to me ! I still have no info on William’s birth or death or of his parents.
You may find it useful to search the database of the Irish Genealogist here on this site, as I notice it does include some articles mentioning a William Fitton. I have not read the articles so do not know whether these pertain to your relation, but at least it is worth checking, if you have not already done so.