Note that Eliza Mary Hunter's marriage took place in a Presbyterian church. Since marriages usually take place in bride's church the Hunters might not have been Church of Ireland as you indicated originally. Therefore, there might be baptisms and early family records that survive. For example, Cairncastle Presbyterian baptisms start 1832. These records will be on microfilm in PRONI, Belfast but I'm not sure if any subscription site has transcripts.
https://www.nidirect.gov.uk/sites/default/files/publications/Guide_to_church_records.pdf
