Sorry about the delay in replying to your query, but have just returned (to Brussels) after two weeks' holiday in Italy.
Some years ago I looked through the old Births, Marriages and Death registers of First Derry Presbyterian Church in Londonderry at the Public Record Office of Northern Ireland (PRONI)in Belfast.I noted down all the Stirlings.
James Stirling married Catharine Goodwin, Fountain Street, Derry at First Derry Church on 4 July 1836.The witnesses were John Clarke and William Tees.They had one son, Henry, who was born on 26 July 1837 at Fountain Street, Derry, and baptised at First Derry on 15 August 1838.James died on 19 January 1838 at Fountain Street.
Catharine Goodwin or Stirling, address given as Wapping, Derry, then married Simon Waddock (I have written his name as Waddock, but it must be my bad writing or I have misinterpreted the writing in the register)of Limerick at First Derry on 23 November 1839.The witnesses were Mary Goodwin and Martha Goodwin McClean (or could be McClure - writing in register uncertain).
(Note:Her name seems to have been spelled in the registers as Catharine - with an a instead of an e - as I have noted it that way twice.)
This all fits in with the information you have, so hope that it adds a few more details for you.
I don't know if James was any relative of my Joseph, who would have been older as his youngest daughter, Elizabeth, was married at First Derry in November 1843.Elizabeth had at least two older sisters, one of whom may have been Caroline, and a brother William, who also became a painter.Stirling is not a very common name in Londonderry and they all lived in the same area of Londonderry, so they may well have been related.Sometimes this is reflected in children's names, but my g g grandmother, Elizabeth Stirling had 14 children and none had the name James.
Hope this helps.Going to N. Ireland in the summer and will let you know if I find out anything else.