SupermanJr,
If the birth Roger has found in 1913 is correct then this is the same family in the 1911 census:
https://www.census.nationalarchives.ie/pages/1911/Antrim/Carrickfergus_Urban/Irish_Quarter_South/192304/
Belfast Telegraph of 4th July 1938 reported Jim Lavery of Irish Quarter South in Carrickfergus helped save the life of a boy - John Wright - who had fallen into the harbour.
Larne Times of 8th Oct 1942 reported that Patrick Lavery of Irish Quarter South had been in Carrickfergus Magistrates Court for disorderly conduct. Fined 2/6. He was in the same paper on 9th Dec 1943, this time in court for riding his bicycle at night with no lights. Quite the recidivist.
Belfast Newsletter of 6th Feb 1930 reported a civil case involving Robert Lavery and his son Charles, both of Irish Quarter South. It hinged around a quantity of fruit that Patrick Collins, fruit merchant, had not been paid for. Court ruled in favour of Mr Collins.
Ballymena Weekly Telegraph of 22nd Feb 1941 reported that Catherine Lavery of Irish Quarter South had been fined 5 shillings for not observing the wartime blackout regulations properly.
Belfast Telegraph 11th April 1940 reported the death of Robert Lavery (wife of Catherine) on 10th April 1940. 24 Irish Quarter South. Burial in Prospect Cemetery, Carrickfergus.
You could ask PRONI to look up the 1939 Register for 24 Irish Quarter South to see who was living there at that time. (The service is free).