The basic idea is to start a separate pod that receives structured logs over TCP and forwards it to Cloud Logging, similar to a locally-running fluentd agent. See below for the steps I used.
(Unfortunately, the logging support that is built into Docker and Kubernetes cannot be used - it just forwards individual lines of text from stdout/stderr as separate log entries which prevents Error Reporting from seeing complete stack traces.)
Create a docker image for a fluentd forwarder using a Dockerfile as follows:
FROM gcr.io/google_containers/fluentd-gcp:1.18
COPY fluentd-forwarder.conf /etc/google-fluentd/google-fluentd.conf
Where fluentd-forwarder.conf contains the following:
<source>
  type forward
  port 24224
</source>
<match **>
  type google_cloud
  buffer_chunk_limit 2M
  buffer_queue_limit 24
  flush_interval 5s
  max_retry_wait 30
  disable_retry_limit
</match>
Then build and push the image:
$ docker build -t gcr.io/###your project id###/fluentd-forwarder:v1 .
$ gcloud docker push gcr.io/###your project id###/fluentd-forwarder:v1
You need a replication controller (fluentd-forwarder-controller.yaml):
apiVersion: v1
kind: ReplicationController
metadata:
  name: fluentd-forwarder
spec:
  replicas: 1
  template:
    metadata:
      name: fluentd-forwarder
      labels:
        app: fluentd-forwarder
    spec:
      containers:
      - name: fluentd-forwarder
        image: gcr.io/###your project id###/fluentd-forwarder:v1
        env:
        - name: FLUENTD_ARGS
          value: -qq
        ports:
        - containerPort: 24224
You also need a service (fluentd-forwarder-service.yaml):
apiVersion: v1
kind: Service
metadata:
  name: fluentd-forwarder
spec:
  selector:
    app: fluentd-forwarder
  ports:
  - protocol: TCP
    port: 24224
Then create the replication controller and service:
$ kubectl create -f fluentd-forwarder-controller.yaml
$ kubectl create -f fluentd-forwarder-service.yaml
Finally, in your application, instead of using 'localhost' and 24224 to connect to the fluentd agent as described on https://cloud.google.com/error-reporting/docs/setting-up-on-compute-engine, use the values of evironment variables FLUENTD_FORWARDER_SERVICE_HOST and FLUENTD_FORWARDER_SERVICE_PORT.