I have been looking for birth registrations for William Fullerton around 1865-66. I have not found any, but there is this possibility
A birth registered in 1865 Galgorm / Balymeena (Antrim) for a William John ALLEN
mother is Eliza Mary FULLERTON
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1865/03565/2313180.pdf(There is a corresponding baptism in Galgorm for this birth. Cannot yet find anything more than a transcription for the baptism.)
This birth interested me for a number of reasons:-
Both the first two names of child
William John (as on 1881 census) and mothers names
Eliza Mary FULLERTON are an exact match.
Also I can see no further children born to these same parents, and no marriage for them.
Also there is a marriage for who I believe is the same William John ALLAN in Ballymena in Nov 1866 to Matilda Foster.
Birth place for this William John born 1865 is Connor. This is the same place that we have as birth place for Jane McGaw and for her son James Fullerton (see 1901 Mearns census)