Francis’s birth c 1845 is well before the start of statutory birth registration. You won’t get a birth cert for him. You may get his baptism, if the records have survived, but you’d need to know where that happened as there is no comprehensive index or database like Scotlandspeople. Likewise Patrick’s marriage. RC marriages were only registered from 1864, non RC from 1.4.1845. Before that you are reliant on parish records.
If Francis’s father was in the RIC then normal policy was to post officers away from the county that they came from. So picking up on your point about the family going to Mayo in 1867, perhaps his father’s family came from there?
Have you researched his father’s RIC records. The records are in Kew with copies in the National Library Dublin. Or you may be able to get some information on Patrick’s origins and next of kin from Jim Herlihy at Blarney Garda station (see contact details on link). Mr Herlihy has written a book on the RIC.
www.policehistory.com/museum.html