Many thanks Dathai and Sinann for the inventive suggestion. I have been through the marriage records with a Richmond Barracks address between 1870 and 1890. In that period, there were over 30 regiments stationed in Richmond Barracks at different times, if the marriage registers are a good guide. In about 20 cases, however, the regiment in question is mentioned only in one record, perhaps indicating that the unit was stationed there for only a short period (this, for example, appears to have been the case for the 104th Fusiliers mentioned by Sinann). Of the units that appear to have been stationed in the Barracks for longer periods the Rifle Brigade shows up in records between 1873 and 1882, the Scots Guards between 1881 and 1889 and the Grenadier Guards between 1882 and 1887. James Coventry's first wife died in 1879. Her death certificate says her husband was a soldier. She died at her family home off Meath Street, not in soldiers accommodation, but it suggests James was stationed in Dublin at the time of her death which may mean he is more likely to have been in the Rifle Brigade rather than either regiment of Guards. This is the best sense I can make of the information though I realise my logic could be full of holes. Unfortunately, neither of James's marriage records list his occupation and I notice that the same is the case for nearly all of the RC marriage records with a Richmond Barracks address. Only the Church of Ireland records mention the groom's regiment.
Rathmore, I found no record for James Coventry in the Chelsea Pensioners records on familysearch nor on findmypast. I got a small number of hits for James Coventrys in the military records on the latter site but none that plausibly connect to my man.