Got this working with this Dockerfile:
FROM bitnami/minio:latest
USER root
RUN groupadd --gid 1001 --system minio
RUN useradd --uid 1001 --gid minio --no-create-home --system minio
RUN chown --recursive minio:minio /data
RUN chown --recursive minio:minio /.mc
USER minio
Bitnami checks for root and handles differently if so: containers/run.sh at main · bitnami/containers · GitHub