OK, just for the record, I've found *a solution*. I don't like it and it does feel hacky, although otoh it doesn't feel too wrong.
Add an extra initContainer with busybox and chown the shared volume with the desired uid and gid. It's ugly but it works. If nobody throws a big reason why I shouldn't be doing this, I'm going to leave it like this.
Thanks for the boosts #hachyderm !