Edit: aghadowey (above) has been more specific on the difficilties but this is what I had ready to post too:That's a complete shame about William's parents names not on the certificate

- I guess his 2nd wife Elizabeth may not have known their names - was she the informant of her death?
Research in Ireland is difficult! The further back the times are, the more so. I have lines back to Ireland that have come to a complete halt at Stranraer and little hope of getting over the divide

You'll need to gather as much information as you can from the Scottish side of things to establish as much as you can about William, every little thing you can find - and that includes about his first wife too, also his religion, his children, an MI if one exists....
He may have been born in Bangor (Down)
He may have been born in Bengour (Cork)
And no doubt there may be other places with a similar name!
The only child I can see to William & Sophia on the IGI are two sons both named James (one christened 1813, the other 1817) presumably the earlier one died and the 2nd was given the same name.
http://shrinkalink.com/60498I wonder if this is the same house - Genoch Lodge, that William was listed at in 1851/1861? Does it look that old a house do you think? (historically Wigtownshire, now under Dumfries Galloway)
Cheers
AMBLY