Hi Trish
Ballinacurra is a village in East Cork. The registration district was Midleton (Middletown). Now that you have a clearer location and date, you could submit an application for a birth certificate to GRO Roscommon. They will do a search for you. If they can't find anything they'll charge you 4 Euros (£3). If they can find the birth certificate, they'll charge you 6 Euros (£5). I appreciate you don't want to throw your money around, but this could be money well-spent. Application form (you have to print it and post it) at
http://www.groireland.ie.
Incidentally, when you are searching for your grandfather, it might be worth omitting the O. A lot of people added or reapplied the O after Independence in 1922, so during the period you are looking for records, the family may not have been using it.
If you go back to the LDS pilot, you'll find there are a few William Briens born in 1881 and registered in Midleton. None quite fit the date, but his birth might have been registered a bit late.
It might also be worth waiting until the 1911 Cork census is released (I'm assuming here that your family were still in Cork at this date) as an important question had to be answered by the women of the household: how many years have you been married? This would help you to pin down the period you need to look in for William and Ellen's marriage. The 1911 census for Cork is expected to be released shortly (but I wouldn't advise holding your breath!).
You might also like to take a note of this webpage in the hope you are able to establish for certain that your family came from Ballincurra:
http://www.igp-web.com/IGPArchives/ire/cork/census/ballinacurra.txt. It contains details from one of those wonderfully rare things in Irish genealogy.... a census from 1842. It could be useful. It contains both Briens and Sheehans, too.
Good luck.