I don't know how this works, but in France and New Caledonia all BDM records are kept at the Town Hall or Mayor's office(Maire).
ASSOCIATION DES MAIRES DE NOUVELLE-CALEDONIE NOUMEA
Activité : ADMINISTRATION TERRITORIALE, PROVINCIALE ET COMMUNALE
CTRE VILLE 32 RUE DU GAL GALLIENI - NOUMEA
BP 4018 - 98846 NOUMEA CEDEX
I have only just found this addrees by accessing the white pages for New Caledonia.
I hope it is correct because I am going to write them to try and get some info re my mother (who was born in Noumea)
my e-mail is firstname.lastname@example.org
- let me know if you get any info