Registration of births started here in Ireland in 1864, so George's birth is far too early to have a cert, instead a search for a possible baptism would be required. There's a likely match for the marriage on the BMD Index so the marriage either took place in a non-Catholic Church, or possibly in a registry Office (Civil Registration of non-Catholic Marriages started in 1845). Do you have a copy of the cert or a copy of the church marriage ?
If you dont have a cert you can use the details on the index to order a research copy from the GRO - (let me know if you need the references) Many of the civil BMD registers are available to view on the free IrishGenealogy website, but these do not include these early marriages yet.
A civil cert or Church of Ireland parish register for the marriage should include some details that could help your search - e.g. occupation of the groom, current residences of the bride and groom, names of their fathers and their occupations, along with names of witnesses - who may be related. The record would also show where the marriage took place, which could help confirm what denomination the couple were..
Based on the attached notes notes it appears that George may have been baptised in St. Mary's Athlone, luckily baptism and marriage records for the parish are available back to the 1760s. Historic registers for the parish are held by the RCB Library in Co. Dublin, and it based on your note also on film in Athlone, and there is also some coverage of the registers for this parish as transcripts on RootsIreland (pay-website)
Military records often include a Parish of Birth (usually civil parish) for those signing up which could help narrow the area to check for baptisms. Church of Ireland records are not currently as well represented online as Catholic registers, so finding the appropriate records can be a problem. There is also the issue of survival of records, many Church of Ireland records were lost in the Civil War.. Based on the latest RCBL list baptism and marriage records for the parish of Booterstown go back to 1824, so unfortunately not far enough to include any details on your George if he was born in this parish.
p.s. Erin Terrace was a terrace of abt 16 houses on Leinster Road, the terrace appears in directory listings around 1847, and later the houses were incororated into Leinster Road, under the revised numbering system Capt. George Stanley lived at 37 Leinster Road.