Hi Quiller,
The consular BMD indices are searchable (for a fee) on
www.findmypast.com. Unfortunately for us genealogists, such registration has never been compulsory, so you may or may not be lucky and find what you are looking for. The birth of my grandmother (b 1915 Argentina) was registered at the British consulate, but those of her sister (Argentina 1917) and cousin (Paraguay 1916) were not.
Bear in mind, also, that consular registration may be significantly later than the birth - sometimes they were done in batches as and when parents got around to it. My gg-grandparents had 7 children in various parts of South America between 1896-1910 - the first three were registered together, another two were registered together later on, and two more escaped registration.
Hope you find what you are looking for

Anna