Each of raspi01 and raspi01.local.berchtold.live is a separate domain.
Unless .local.berchtold.live is your local search domain, DNS clients would have no way to arrive at the second when trying to resolve the first.
Your container running Pi-hole doesn't seem to be aware of this, but since you are using Pi-hole as DHCP server, I see you've correctly configured that as the local domain - but your configuration is currently missing an expand-hosts option.
Please see [SOLVED] How can I get the local domain name auto suffixed to hostnames in DHCP? - #2 by Bucking_Horn for ways to address this.