As the source given was Fife Council, Lair Regist. I reckon you would need to get in touch with Fife Council.Unfortunately I've not been able to follow up the possible death of James 1 so far.
I thought I'd asked at the time how i would do this (it doesn't appear on SP ) but i cant see it now - how do I go about that?
Maybe try:
http://www.fife.gov.uk/fife_council/ind ... 0B384B8758
Probably just the date of burial, but what you may find is that there is mention of others in the same lair, so you may find that Christina/Christian is buried there too. There will be no mention of parents or witnesses I wouldn't think - it will be a record of burial only.As its before 1855 what level of information would I get ? Would I get His parents, witness etc?
There are no MIs in the Fife books that I can see for either James Gourlay or Christian in Kilconquhar. (I take it you are aware of the local pronunciation for the place?)
The original images for the census can be found on ScotlandsPeople, if you do not already have them.
Best wishes
Lesley