Hello Pamela, I live about 14 miles from Ardee but we have no volunteer there. That said I am a member of the County Louth Archaeological & History Society and they have been in existence since 1903, and while there is no mention of Dr Goulding the Ardee Dispensary Minute books from 1813 to 1851 have been reproduced and are available to view on J Stor (online library of academic journals and can be viewed or downloaded there. There is free access to members. https://clahs.ie/ For some odd reason I have downloaded that article myself but not read it in detail.
There is a very good Louth website here http://jbhall.clahs.ie/ Google Louth Genealogical Sources if you cannot use the link. There are only 2 Goulding hits and not yours.
The library and archives of Louth may be of assistance here https://www.louthcoco.ie/en/services/library/reference-local-history-library/ there is also a genealogy section and the archive is here https://www.louthcoco.ie/en/services/archives/
The name Brannigan and Branigan are both found here in Louth, Goulding however is not common here.
The National Library have digital photos if you search Ardee however none of the medical centre. https://www.nli.ie/digital-photographs.aspx
The tragic death may be in the local paper and oddly for historical reasons the best subscription site is the British Newpaper Archive for this period. https://www.britishnewspaperarchive.co.uk/ Also the physcians society may have something. https://www.rcpi.ie/
I do not know the origin of the name Brannigans Cross but probably Brannigans lived there at some point or owned the land there, it is still viewable on Google Street view between Collon and Monasterboice a few miles from Ardee, funshog is on the Ardee to Collon Road.
There is also a Facebook page on Louth Genealogy and someone there may be local to the area, there is also a small Ardee History group on Facebook and this appears to be an open group.
Will search for children now but saving this as I often lose the content when I leave.