Burial Plots - records from 19th C

I'm trying to solve a 19th century riddle of the relationship of a small number of people in the same townland with the same unusual surname. It occurred to me that if I could identify groupings in burial plots, this would greatly help.
In the local RC graveyard, I know that there are several burials in the 20th C that do not have a headstone, or the headstone is now obliterated. These were subsistence farmers, so I think the earlier ancestors may also be buried there with no headstone. I certainly can't find them.
Do RC churches keep records of the actual plots from 19th C? Would they be open to general research? It's not a big cemetery, and I'd love to leaf through all records looking for the surname. Is that a pipe dream?
I sent a general email to a contact address listed on the diocesan website, but I'm not even sure what I'm asking for. An index of who is in which plots, I guess.
I'm not a church-goer, is it appropriate to hang around after a service and approach the local priest? I did go into this small rural church on a weekday, but nobody was around.
Or should I be approaching the bishop's office? Sorry - I have no real clue of how church administration works!