@Romain M Thank you for the additional context.
You should be able to set up URL rewrite on the CDN to route requests to BlobStorageName to automatically append a .json extension.
I have a created a sample regex here. Please test it, modify according to your use case and use it to configure the URL rewrite.
The URL https://customDomainName/BlobStorageName/4 should rewrite the path to https://customDomainName/BlobStorageName/4.json
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