Accessing API Gateway both Public and Private endpoint

My customer want to access both private and public REST API Gateway from one VPC with enable private DNS name. This VPC has VPC endpoint to access private API Gateway and internet GW or NAT GW to access public API Gateway. Public API Gateway assumes an external service that is built on AWS, not my customer's service.

According to the official AWS document "https://docs.aws.amazon.com/apigateway/latest/developerguide/apigateway-private-apis.html", if I want to public API Gateway in these situations, they need an edge optimized custom domain.

Is it possible not only for edge-optimized custom domains but also regional custom domains?