I had a similar problem just a few weeks ago and was hoping to visit my 3 x great grand parents grave whilst visiting Scotland from Australia, (A different part of Scotland, but I'm sure they will help in all towns if they can) I contacted the bereavement services with the full names, dates of deaths, places of death.
I had a very nice reply stating that my 3x Gr Grand father who had died in 1881 was buried in the old cemetery which was now closed. but they had located his wife's grave, not only did they arrange to meet me at the cemetery, but also took me to the grave which was lovely, I did have a bonus though and found that their Daughter & their Granddaughter were also buried in the same grave.
Again this was a very poor family 16 children & only 3 living to adulthood, so of course no headstone. I found a stone mason near the cemetery when I went to buy.
flowers for the grave and ordered a modest one, this is due to be placed in the next few days, and it is a lovely feeling knowing that anyone walking around the cemetery will know the family did have someone.
It is worth a try.
Good luck
Triish