I managed to have some luck with this. My code looks similar to yours OP, but my compose entry looks a bit different, and I wonder if your issue was simply related to network: host. Shouldn't it be network_mode: host and outside of build?

services:
  avahi:
    build:
      context: ./config/avahi
      dockerfile: DockerFile
    network_mode: host
    cap_add:
      - NET_ADMIN
      - NET_BROADCAST

On another note, I'm running into hostname conflicts between Avahi in the container and the host. Need to confirm, but setting disable-publishing=yes in the host avahi-daemon.conf is showing promise, in case it helps anyone else and is suitable for your use case.