custom domain names with private cluster with ExternalDNS

[kangsheng89]

Refer to documentation:
To register custom domain names to your DNS provider, we recommend using ExternalDNS. AWS Gateway API Controller supports ExternalDNS integration through CRD source - the controller will manage DNSEndpoint resource for you.

The issue when we use private cluster without internet access, we are unable to access to Route53 Api from VPC private link. Is there a better method that can support custom domain name with ExternalDNS on private cluster???