There seems to be a bit of confusion over Bannockburn 'Burial Ground'
The Commonwealth War Graves Commission named St. Ninian's Churchyard as Bannockburn Parish Burial Ground which led anyone seeking the burial place (or commemoration on family gravestones) of any service personnel killed/died in WW1 to go searching in Bannockburn Cemetery.
Bannockburn Cemetery is a completely different place - it also has CWGC burials and not very far from St. Ninian's.
Stirling Archives would be the best place to enquire as (I think) they hold the burial registers for the Cemetery. From memory, St. Ninian's Churchyard's records are few and far between.
https://my.stirling.gov.uk/libraries-archives/archives/accessing-archives/Anne

P.S. They would also be able to tell you when the first burial in the Cemetery was.