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In my experience of expatriate family research, consular registration of births has been far more common than consular registration of deaths. The reason for this is likely be that a consular birth certificate was regarded as useful for the child’s future in establishing citizenship etc, whereas there is no such benefit at death. A further point is that the likelihood of using the consular registration route often depends upon how developed the local country’s civil registration arrangements were.
You might do better to name the country or countries concerned, so that those with local research experience can suggest avenues for further enquiry.
I’m confused by your saying that you know the birth was registered at the Consulate but have had no luck in ordering the certificate. What has been the difficulty - have you not found the index details (but if so how do you know it was registered?) or have you found the details but been unable to place an order?