My recollection is that the Latrobe State Library in Swanston St holds the NZ BDM on microfiche. Plus they should have the 'District Keys' books which translate each entry's folio number to a place of registration.
IF you pinpoint the births of all the other children in the family, then confirm the place of registration (which may be different to the place of birth but is usually in the same district); that should narrow the time period down even more.
Did a birth registration search a few years ago for a guy born in the late 1950s near Wanganui - he was #5 in a family of 9. The four before and after him were registered. Turns out his mum was ill after his birth, they lived up the river, his dad had to go back to his forestry work while grandma and aunty looked after the family. I guess registering his birth was simply in the too difficult at present basket, then never got done. But he wanted a passport to travel to a family wedding - all made extremely difficult by no birth registration. We got there but it was close.