Rootsireland.ie (PPV website) have the records for St Pauls Arran Quay. It is showing 7 children baptised to a Thomas Kiernan and Catherine Murray:
John 1844
Christopher 1850
Anne 1853
Mary 1855
Thomas 1858
Peter 1863
Alicia 1869
Hard to tell if this is all the one family as 1844 - 1869 is fairly stretched out, though not impossible.
That looks like the Peter and Alice that you mention and the Mary looks spot on as well for Mary Anne. You would need to purchase credit to see the full results to be sure. They are transcriptions of the records rather than images of the records themselves.
EDIT to add.
The above came up on a search just for St Pauls. it you do an All Ireland search you get 2 more baptisms
John 1845 Co. Meath
Michael 1847 Co Meath
(Co Meath boarders Co Dublin) Anyway the 2 extra children may well indicate this is the one family as sorts of fills the gap between John 1844 and Christopher 1850.
Since there is a second John in '45 - it would indicate that the first John born '44 died in infancy and sure enough Glasnevin Cemetary have a record for 'John Kiernan - died aged 4 weeks' buried there alone in 1844.
http://www.glasnevintrust.ie/genealogy/ I'm afraid that is another PPV site.
The only one that will have a Civil Registration Birth Cert will be Alice/Alicia - if you are interested in getting that, then that should be easy enough to source.
Anyway, a few leads to continue your search (if expensive ones) !!!
