Nothing else other than this one stone, sorry. The records begin 1861 for the Walled Burial Ground and for the ancient churchyard. No sign of Kate McCarter/McArthur in the list of monumental inscriptions.
The 3 girls seem to be the only occupants of plot No.8 in the Walled Ground. This is an early number and the plot is against the wall, so it would probably have been purchased before 1870, which is the date of the 1st interment. The records show the following - (all children of William Dick & Marion Calderhead)
MARION DICK, Stane, aged 2 months. Buried 25th February 1870.
JESSIE DICK, Edinburgh, aged 6 years. Buried 21st January 1880.
CATHERINE DICK, Portobello, aged 21 years. Buried 5th March 1882.
Stane is now part of the modern-day town of Shotts. Portobello is a lovely part of Edinburgh on the firth of Forth.
There was a William Calderhead, Inspector of Poor at Wishaw in 1907, I wonder if he is related to this family?
Good luck with your research.