Peggy,
The tithe applotment records for 1827 list a James Happer (plus some other Happers) in Drumadd townland, which is in the parish of Tamlaght. This is not too far from Cookstown:
https://cotyroneireland.com/tithe/tamlaght.html
The names all appear twice which suggest they have been duplicated.
By the time of Griffiths Valuation in 1859 there are no Happer households listed there. There was a Robert Ferguson (plot 9) on a 21 acre farm. Ferguson is a common name in the area though and it may just be a coincidence.
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=nameS…
Registration of births & deaths didn’t start in Ireland till 1864. Prior to that you are heavily reliant on church records. You didn’t say what precise denomination(s) the families were (save that Mary married a Catholic). Drumad(d) is in the parish of Tamlaght. If Church of Ireland, Tamlaght parish has baptisms from 1801, marriages from 1829 and burials from 1834. If Presbyterian, then Coagh Presbyterian church has baptisms from 1839 and marriages 1820-22 and 1845 onwards. Presbyterians generally don’t keep burial records. There’s copies of these records in PRONI (the public record office) in Belfast.
If the family came from Tamlaght and were Church of Ireland you may find Mary & Bettie’s baptism in the Tamlaght records. You might also find James Happer’s burial in 1843, plus the baptisms of any children they had. But if they attended a different church in that parish, it appears that no records exist for the years you need.
With regard to the Fergusons and a castle, I am not aware of any significant castle in Tyrone that was still inhabited in the late 1700s and associated with the name Ferguson. There is Killymoon Castle near Cookstown (rebuilt in 1802 after the original burned down) but it belonged to the Stewart family.
https://www.midulstercouncil.org/visitor/places-to-visit/historic-house…
There is of course Castle Caulfield, not far from Cookstown, but it belonged to the Caulfield family and was in ruins in the mid 1600s.
If Bettie Ferguson’s father was a Lord, then that normally indicates he was of a noble family (ie Baron, Viscount, Earl etc). They are all listed in Burke’s Peerage and Debrett. You can search them on-line (subscription sites). Some major libraries also keep copies.
The lack of records in Ireland makes researching in the 1700s very difficult. It may be hard to substantiate the story about the Ferguson family’s background. Whether an elopement in the 1700s would have made the newspapers is debatable.
Sometimes family stories get a little exaggerated too. It’s surprising how many young men and women from a family in the nobility reportedly eloped from a castle with a servant, (often leaving an unclaimed fortune or were disinherited). They often stowed away on a ship to America. Sometimes their castle was reportedly taken from them by the English. These themes recur with remarkable regularity.