The Catholic Chapel at Phibsborough was part of the parish of St. Paul's based at Arran quay in the city, so that would be the parish to check for possible records. The parish has baptism and marriage records going back to 1730s with a few gaps.
I did not find a promising match for Patrick jnr's baptism in the available Catholic records - did he live to the 1901 census ? do you have a county of birth for him ?
edit : I think I see your Patrick in 1901 - cattle exporter, in Charleville Rd, Glasnevin , with wife Lucy - Patrick born "Co. Dublin", Lucy in Co. Tyrone. They married in Rathmines in South Dublin...unfortunately the parish records for Rathmines for this timeframe dont give residences of the parents.