Assign FQDN for Internal Services in a Private Kubernetes Cluster
Currently, there is no ready solution for this.
Only thing comes to my mind is to use MutatingAdmissionWebhook. It would need catch moment, when new Kubernetes service was created and then modify ConfigMap for CoreDNS as it's described in CoreDNS documentation.
After that, you would need to reload CoreDNS configuration to apply new configuration from ConfigMap. To achieve that, you can use reload plugin for CoreDNS. More details about this plugin can be found here.
Instead of above you can consider using sidecarContainer for CoreDNS, which will send SIGUSR1 signal to CoreDNS conatiner.Example of this method can be found in this Github thread.