Alpine Docker: hostname service wont start: "sethostname: Operation not permitted"

Question

I created a simple image based on Alpine which uses openrc. When trying to start the hostname (or netowrking) service with service start hostname, the following error occurs:

 * Setting hostname ...
hostname: sethostname: Operation not permitted                                                                                                                                                                                                                          [ !! ]
 * ERROR: hostname failed to start

The hostname of the container is set and is correct inside /etc/hostname

Found this post, however I'd prefer not to add SYS_ADMIN cap: Alpine Linux docker set hostname

What is needed to resolve the problem?

Snippet from my Dockerfile:

FROM alpine

VOLUME /sys/fs/cgroup

RUN apk add gitea openrc postgresql15

COPY entrypoint.sh /entrypoint.sh

RUN openrc &&\
    touch /run/openrc/softlevel &&\
    chmod +x /entrypoint.sh

CMD ["/bin/sh","/entrypoint.sh"]