... I found records for some of this branch of my tree on the Catholic Registers at the NLI which was a huge surprise as the family are Church of Ireland.
Just when you say that, note that Henry McLaughlin was baptised in St. Peter's RC Church, Belfast, on 1 February 1893, I have only seen a transcript and not the original register entry, but the transcript leaves no doubt that it's him, for the recorded address was 114 Servia Street. What is slightly odd is that Rose was recorded in her married name of McLaughlin, but no father was stated. The sponsor was an Agnes Dugan.
I'm ultimately not sure that James and Rose and son Henry are connected, they certainly tick boxes, but that might just be a coincidence. I'm not aware of Henry Dugan and Maria Wills having a daughter Rose, though I suppose she could be a niece. It's frustrating not being able to find a marriage for James and Rose, James' omission in that baptismal transcript is making me wonder if there was one.