How to allow a non-root user to write to a mounted EFS in EKS
If anyone comes across this later I resolved my issue by using an initContainer for any Pod that needed to write to the file system.
For example:
apiVersion: v1
kind: Pod
metadata:
  name: app3
spec:
  containers:
  - name: app3
    image: busybox
    command: [&quot;/bin/sh&quot;]
    args: [&quot;-c&quot;, &quot;while true; do echo $(date -u) &gt;&gt; /data/out3.txt; sleep 5; done&quot;]
    volumeMounts:
    - name: persistent-storage
      mountPath: /data
    securityContext:
      runAsGroup: 1337
  initContainers:
  - name: fs-owner-change
    image: busybox
    command:
    - chown
    - &quot;root:1337&quot;
    - &quot;/efs-fs&quot;
    volumeMounts
    - mountPath: /efs-fs
      name: cog-data
  securityContext:
    fsGroup: 1337
  volumes:
  - name: persistent-storage
    persistentVolumeClaim:
      claimName: efs-claim

The rest of the definitions match what was in my question.