Your customer wants to have a single CloudFront Distribution with multiple domains/subdomains and be able to serve that content using HTTPS, correct? Would all the content be served from a static set of origins or does the origin change based on the domain name?
This configuration is possible, your customer can configure the CloudFront distribution with multiple CNAMEs and use AWS Certificate Manager to generate a public certificate with multiple domain names(SAN).
From the example that you've shared, your customer would still have to use Lambda@Edge to change uri of the request, depending on the host header, as it seems that according to the host, the uri can change.
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