Here is the Trove newspaper cutting for the death announcement in 1876
https://trove.nla.gov.au/newspaper/article/107179625 Evening News 8 July 1876
On reading that cutting I offer that her death 'a short while ago' was in May or June of that year and was in NSW. So there's some reasons for not finding her death indexed at the online NSW BDM. Firstly the index was not in existence until a team of volunteers transcribed the records in the 1930s, so the Registrar General's copy if it existed would have been received as part of a quarterly return from the local deputy Reg Gen officer for perhaps the Bathurst district, back in Oct 1876. IF (and this is a possible) that quarterly return was a) not sent b) not received c) too difficult to transcribe at the time then it has not ever been actioned by the clerks in the then 1876 Reg Gen's office
HOWEVER, NSW BDM death certs are quite detailed, and even in 1876 they included the equivalent of the burial order, so were lodged locally with the deputy registrar by the undertaker after he had obtained family info, the medico's signature confirming the deceased was deceased and causes etc.
So a search may be needed of all the many cemeteries in the Bathurst district .... an extensive district then and still is now...
If the death was sudden and she had not been under the care of a medico for several weeks, then the police magistrate in that district would have needed to consider if there were suspicious circumstances and involve a coroner or at least a Justice of the Peace.... But a burial order may have been issued at that time, and BEFORE any coroner's enquiry happened. IF THAT IS THE CASE .... well, in many instances in that era ... once the undertaker had a burial order, he proceeded with the burial and then he had no further need to remember to do any further paperwork, for it was often the case that the Police Magistrate or at least the clerk of petty sessions, or the sheriff or their spouse was the part time deputy Registrar.... sparse populations spread over huge territories, not many BDM events, so not much practised knowledge on handling the bdm paperwork for the Sydney Registrar General who was also responsible for Land Titles, Deeds, Land Sales, Conditional Purchases/leases etc etc etc.
So it is entirely possible that her death was registered and is mis-transcribed, or was registered locally and not adequately transmitted to Sydney HQ, or has been lost in the EDP system when NSW BDM first computerised back in the 1970s and 80s, and recalled all the non metropolitian registers to HQ.
There are other clerical admin issues, but those seem to me to be the significant ones, assuming she died in the Bathurst BDM District ... similar thoughts though for other non Metropolitian BDM districts with infrequent need to register bdm events, thus part timers responsible for the clerical recordings.
JM