St. Paul's parish is located in the west of Dublin North city, north of the river Liffey whereas St. Andrew's is in the south east of the city and south of the river Liffey so not adjacent - with several parishes in between. In both cases there are Catholic and Church of Ireland parishes with these names. Just to complicate thing a little further there is also a civil and Church of Ireland parish of St. Anne's (St Annis. ?), which is located adjacent to St. Andrews civil parish.
Have you ruled out the baptism in St. Andrew's RC of Pat Gorman 4th May 1795 with parents Peter and Suzanne (no maiden surname given) ? .. maybe your Patrick was a little under age to join the army and pushed his year of birth back when joining ...
Pity that his military record does not mention denomination, they sometime include this detail... I presume the Manchester Cathedral mentioned is Church of England ?
Have you found baptisms of his children - what denomination were these ?
If that 1895 baptism is not your Patrick then my feeling is that either your Patrick was Church of Ireland and therefore baptism record lost, or baptism was Catholic and recorded on a page now missing or damaged, or the information he gave when enlisting was inaccurate, with regard to either his age or place of birth, or both. Discounting some or all of the enlistment details creates a problem as there's no additional clues to work with at this point... so any records found after this become 'possible' matches with no straightforward way to prove or rule out. Getting back this far (i.e. Ireland before 1800) is doing very well as records do not give much detail and many people struggle to get back past the 1830s.