Trying to be helpful Harper and Harpur are both common surnames. Is it clearly one spelling only used throughout in the family bible.
Where was the family bible located when first used by yourselves in a Harper household? Are you sure it would not be easy to trace backwards from the USA etc ? I notice there is a gap between the family children after the twins were born and the next child Jane, which is natural, but the timing is also when the major irish famine occurs, and I think family search.org has a special emigration list of those who left between about 1846-1851.
Child 2.James E. Harper born 2/14/1836 in Derry You have ruled him out, but his birthplace remains, was Derry named in the family bible ? Are you aware Derry or Co Londonderry as it was known, is the adjoining county to Co Antrim? Randalstown is close to the border of the the two counties. There is no Derry townland or civil parish in CO Antrim so that means the family must have moved counties after his birth or Jane went back to her mothers to have the child. There is no trace of McCulloughs in Griffiths Co Londonderry.But there are other spelling
In 1859 in Griffiths valuation see askaboutireland.com under Co Londonderry are three Harper households [and a lot more Harpurs].
Errigal Harper James Tamnymore
Banagher Harper Jane Templemoyle
Lissan Harper William Ballybriest
One can do the same for Co Antrim
I don’t have the time to search exhaustively for named Harpers using
www.familysearch.org I noticed an Andrew Harper having children with his wife Elizabeth Palmer with a christening and births registered as Crumlin Co Antrim Crumlin is in Camlin civil parish. In 1826 there were no Harpurs in Camlin civil parish, but there were nearly 20 Palmers So where did he come from?
Going back to earlier surviving bits of census
http://www.census.nationalarchives.ie living in 1831 at
5 in Ballymughan More (Magherafelt, Loughinsholin, Londonderry) census Forename Number of house Number of families in each Number of males in family Number of females in family Number of male servants Number of female servants Number in Established Church Number of Roman Catholics Number of Presbyterians Number of other Protestants
McCullagh Jane 5 1 1 3 - -
If one was Jane McCullough ,then one of the local churches was
1st Magherafelt Presbyterian church with the marriage date missing!
Baptisms, 1771-81(with index), and 1813-1964;
marriages, 1769-82 (with index), 1819-28 and 1845-
1963; marriage notice books for Magherafelt
Presbytery, 1845-1946; session minutes, 1703-82 and
1818-56; committee minutes, 1848, 1852-3 and1911-
45; list of elders in 1828; ministers’ visitation books,
1823-32 and 1841; ministers’ diaries, 1833-54;
ministers’ account book, 1889-1904; history of the
congregation prepared in 1853.
[Baptisms 1703-06 for Dawson's Bridge may include
baptisms for 1st Magherafelt – see under
P. Castledawson above.]
CR3/13; D2725/6; D1926/1-
2; MIC1P/450; MIC1P/460
So thats as far as I can go, good luck onwards JimG