RabbitMQ Cluster Kubernetes Operator on minikube

As I alread suggested in comments, you can solve it running:

minikube ssh -- sudo chmod g+w /mnt/app/rabbitmq/ 

Answering to your question:

Is there a way I can add that to my manifest file rather than having to do it manually?

You can override the rabbitmq statefulset manifest fields to change last line in initContainer command script from chgrp 999 /var/lib/rabbitmq/mnesia/ to this: chown 999:999 /var/lib/rabbitmq/mnesia/.

Complete RabbitmqCluster yaml manifest looks like following:

apiVersion: rabbitmq.com/v1beta1
kind: RabbitmqCluster
metadata:
  name: rabbitmq
spec:
  replicas: 1
  service:
    type: NodePort
  persistence:
    storageClassName: local-storage
    storage: 20Gi
  override:
    statefulSet:
      spec:
        template:
          spec:
            containers: []
            initContainers:
            - name: setup-container
              command:
              - sh
              - -c
              - cp /tmp/rabbitmq/rabbitmq.conf /etc/rabbitmq/rabbitmq.conf && chown 999:999
                /etc/rabbitmq/rabbitmq.conf && echo '' >> /etc/rabbitmq/rabbitmq.conf ; cp /tmp/rabbitmq/advanced.config
                /etc/rabbitmq/advanced.config && chown 999:999 /etc/rabbitmq/advanced.config
                ; cp /tmp/rabbitmq/rabbitmq-env.conf /etc/rabbitmq/rabbitmq-env.conf && chown
                999:999 /etc/rabbitmq/rabbitmq-env.conf ; cp /tmp/erlang-cookie-secret/.erlang.cookie
                /var/lib/rabbitmq/.erlang.cookie && chown 999:999 /var/lib/rabbitmq/.erlang.cookie
                && chmod 600 /var/lib/rabbitmq/.erlang.cookie ; cp /tmp/rabbitmq-plugins/enabled_plugins
                /etc/rabbitmq/enabled_plugins && chown 999:999 /etc/rabbitmq/enabled_plugins
                ; chown 999:999 /var/lib/rabbitmq/mnesia/ # <- CHANGED THIS LINE