Hello everyone. I see it's been a little while since the original post by Bee29, but I've been doing a lot of research into this section of the Beckett family and can possibly shed some light. Open to correction.
Archibald Beckett (b circa 1822, d circa 1891/2) was the eldest of eleven children of Michael and Mary Beckett (or Becket) of Turlough. He married Margaret Bourke and they had three children: Isaac (1864), Ann Jane (who married a James Acton or Picton) and John (1870).
The Isaac Beckett who moved to NZ I believe was on a different line. He was the third child (of 13) of James (c 1822) and Elizabeth (nee McAdam c 1828) Beckett. 'NZ Isaac' (1852-1920) married Clara Wheeler and they had six children: James, Letitia, Agnes, Priscilla, Mathilda and Lizzie.
While in NZ, Isaac's nephew William (youngest son of James and Elizabeth's 6th child, William) wrote to him and asked if he could come and visit. Isaac declined. But Isaac's daughter Letitia privately wrote to her cousin entreating him to come, which he duly did, and the pair eloped. They had seven children: Patricia, Evelyn, Neil, and four others (still living).
James (married to Elizabeth McAdam) was the son of Isaac and Mary (possibly Lindrum) Beckett. He had a brother, Alexander, who married Elizabeth Naylor in 1846.
This older Isaac had a brother, John, born before 1738/9 in Castlebar (died 9 Mar 1824 in Turlough). John married a Mary Johnson, and they had five children: Nathaniel, James, Alexander, Thomas and Jacob. Most of this branch ended up at Beckett's Creek, Ontario, Canada.
I have no proof as of yet, but a sneaking suspicion Archibald's father, Michael, may have been a brother of Alexander and James. There may also have been another brother, Isaac (from whom Jacob Duke Beckett of Spencer Park descends) and another called William. All unconfirmed speculation I'm afraid, but if you can't prove it, you can't disprove it either