Our working hypothesis is that this is an oncolite.
These can be deposited by algae, and would be consistent with the area where you have found them.
There was a large paleo-lake, Lake Manuherikia, in the area, and these may have been deposited in the shoreline waters of the old lake.
The excerpt below is from an report noting:
‘Ostracods, algae (including oncolites), fish and bird remains are dispersed within shoreline facies.’
Lignite Resources of Central Otago by B.J. Douglas 1986 pg ii