Kubernetes: Have no access from EKS pod to RDS Postgres
Have you tried to enable "dns propagation" in the peering connection? It looks like you are not getting the internally routable dns. You can enable it by going into the setting for the peering connection and checking the box for dns propagation. I generally do this will all of the peering connections that I control.
The answer I provided here may actually apply to your case, too.
It is about using Services without selectors. Look also into ExternalName Services.