The EAP Operator creates two services to access the statefulset pods.

One service named "xxx-headless" is used to coordinated pods scale down.
The other service named "xxx-loadbalancer" is used to access the pods during regular operation. This service is created with the Type set to "LoadBalancer".

This is not correct. The LoadBalancer type is meant to "Expose the Service externally using a cloud provider's load balancer." (https://kubernetes.io/docs/concepts/services-networking/service/).
Instead this service should be created with a type of "ClusterIP" to "Exposes the Service on a cluster-internal IP. Choosing this value makes the Service only reachable from within the cluster."

This service was not intended to be reachable from outside the cluster. The Route resource is used for that purpose.