Just some more info on the Churches you mentioned.
St. Patricks Catholic Church, Cowgate, Edinburgh.
Canongate Kirk, Canongate, Edinburgh.
If you " google " these churches you will find various sites which give a history and details.
Both are within walking distance of each other in the Old Town part of Edinburgh.
As far as I am aware the Catholic only Cemetery - Mount Vernon - did not open until 1895.
The Catholic Archives say that their online records are available to view at Scotlands People. Back to them again for viewing Scottish records. Perhaps you would like to contact them to confirm this.
Hope this helps you.
Dorrie