Hi Jen,
He's a fine looking gentleman. Aristocratic?
Yes, I think you're right. Probably only distant connections at best.
A lot of the bdm records were lost when they were centralised and the archives building was destroyed. However, I have managed to find the civil marriage record (Aughnacloy (Aghnacloo/Aghnacloe) /Carnteel Parish) for my relatives, John and Margaret. It seems that Catholic parish names can be different and there are sometimes multiple spellings. I have found this an added confusion.
I found this list of flax growers in 1796 for Co Tyrone... my relatives line of work... but you never know, it may be useful for tracing your William's father if this occupation makes sense in your family's context:
Walker David Donaghedy Tyrone
Walker Eleanor Clogher Tyrone
Walker Greer Clonfeacle Tyrone
Walker James Aghaloo Tyrone
Walker James Ardstraw Tyrone
Walker James Cappagh Tyrone
Walker James Clonfeacle Tyrone
Walker James Tamlaght Tyrone
Walker John Aghaloo Tyrone
Walker John Ardstraw Tyrone
Walker John Donaghedy Tyrone
Walker John Errigal Keerogue Tyrone
Walker Joseph Donaghedy Tyrone
Walker Lindsey Clonfeacle Tyrone
Walker Martin Errigal Keerogue Tyrone
Walker Robert Arboe Tyrone
Walker Robert Clonfeacle Tyrone
Walker Robert Dromore Tyrone
Walker Robert Errigal Keerogue Tyrone
Walker Robert Tullyniskan Tyrone
Walker Samuel Donacavey Tyrone
Walker Thomas Clogher Tyrone
Walker Thomas Donaghedy Tyrone
Walker Thomas Killyman Tyrone
Walker William Clogher Tyrone
Walker William Clonfeacle Tyrone
Walker William Tamlaght Tyrone
The Irish Family History Foundation pay as you go searches may be worth considering:
http://www.irish-roots.ie/Good luck with it all
Julie