Good to hear this is an active thread.
I gather now that "West Church burial grounds" refers to the graveyard at St Cuthbert's (at the bottom of Lothian Road).
As a general query, how were the burial arrangements handled by poor RC folk in Edinburgh mid 19th century?
The people I'm researching were all illiterate and death was an all too frequent visitor on their households (round about the Cowgate). I'm guessing that there were unmarked sections of the graveyards at St Cuthbert's and Calton Hill for those who couldn't afford headstones etc and weren't in the Church of Scotland.
The St Cuthbert's website
http://www.st-cuthberts.net/famhist.php#graves refers us to the Recorder of District Burial Grounds at Mortonhall Cremetorium who hold the chronological register of burials at St Cuthbert's from 1804 to the present day. It is just a diary, so you need to make an appointment armed with exact dates by the sound of it.
Gerry