carolnotjoyce Stillbirth is a term which has a specific legal meaning, which in Britain is that the child was not born alive, however briefly, at the time of it's birth, and at that time period, that the pregnancy had lasted for at least 28 weeks by the time of the birth.
If your sister's birth circumstances met those criteria, then it would have been a legal requirement that her birth would have had to be registered in the stillbirths register.
However, when ordinary people speak about something they don't usually use words and phrases with regard to their specific and precise legal meaning, and I can imagine that many lay people would use the term stillbirth to refer to a child who was born alive but who only lived very briefly.
If that had happened in the case of your sister, then her birth might perhaps have been a live birth, and if it was then her birth would have been registered as a live birth, and even if she was unnamed in regard to her forename, her forename, in the birth registration index ledger, would be listed as female, her surname would have been listed as her father's surname, and your mother's maiden surname would also be listed in the birth registration index record.
If her birth was registered as a live birth, then she would of course, also have a death registration.
When the GRO official failed to find a birth record in the stillbirths register, did they then double check that the birth hadn't been registered as a live birth, or did they just assume that your information was wrong, and not check any further, who knows, but perhaps you might know. ?
If her birth wasn't registered in the stillbirths or the live births registers, then the only other possibility would seem to be that the pregnancy didn't last for 28 weeks.
Presumably no mention of any funeral arrangements was made.
If your late mum's medical records go back that far, and if you were permitted to access them, the the birth would probably be mentioned in them.