If you just want to call the API with multiple names, there is no need for second distribution but you could add alternative names to your "api" distribution. https://docs.aws.amazon.com/AmazonCloudFront/latest/DeveloperGuide/CNAMEs.html
I'm not sure how good idea is to have 2 levels of caching but you could configure "api" distribution as custom origin for "project" distributions. I recall there might be some loop-detection in Cloudfront not allowing this, but should be able to work-a-round by limiting origin request headers you forward from "project" distribution to "api" distribution.
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