I do know Larne quite well.
The two main cemeteries in Larne (Greenland or New Cemetery and McGarel) should be covered by council records apart from the small R.C. section of McGarel, the records for which have been lost.
I think that I am correct in saying that all Churches in Larne Town with attached burying grounds are included in the book to which I referred. The one with most burials is St Cedma’s (Church of Ireland) formerly known as Inver parish. The Non-Subscribing Presbyterian Church also has some burials but not many. The other church graveyards have only ones and twos; many have none. The book also includes two cemeteries at Kilwaughter which is about 3 miles outside Larne.
Then once you move outside of the town you have a cemetery attached to St. John’s Glynn (Church of Ireland) – about 2 miles from Larne. There are also Presbyterian and Church of Ireland Cemeteries at Cairncastle which is about 5 miles from Larne….and so on. None of these are in the book.
It is hard to know what to advise. Burial records for St. Cedma’s are available at PRONI but will only be relevant if he was buried without a headstone. I was through them last week but can look at them on my next visit (nothing planned in the immediate future). If William was Church of Ireland then the Glynn Church would be worth a visit. I can do that for you when the weather clears a bit.