Chris,I found 4 births for Patrick in Galway;1844,1845,1849 and 1854.None come up with the father's name as Timothy but that is not unusual as many certs didn't have parents' names on them.For his father Timothy I found this; Marriage: 10 February 1844 parish:Dunmore county:Galway groom:Timothy Larner religion:Roman Catholic bride:Mary Cosgrove religion:Roman Catholic witnesses:John Keane/Judith Kirrane
I did find another couple of things but am not 100% sure they are your family but worth looking at.
Royal Irish Constabulary: Timothy Larner age:19 birthplace:Galway marriage:Y enlisted:1847-1866
I found 2 births of children born to Timothy Larner in Galway; Bridget 1839 and Margaret 1840.As they were born before Timothy's marriage,Timothy may have been married before.
Also,a Timothy Larner appears in the 1911 Irish census as a 73 yr old man living with a son Martin and four grandchildren.If they got his age wrong it could be Timothy but if not it isn't or he would have married at 5.I would suggest if you have an LDS library near you,go to it and check the parish registers to see if you can find the Larners.The name was also spelt Lardner.Hope this helps,Kim