Use Siri - probably HomeKit - does not connect

Hello everybody,

I am new to Home Assistant. Also new to Automation with Apple iOS devices. Not new to computers.

My aim is to use Siri on my Watch and/or my iPhone to run automations in HomeAssistant. My understanding is, I need to install the HomeKit Bridge, and this is, what won’t work.

My Setup:
Home Assistant is installed in a Docker container on a Raspberry Pi running Bullseye, using the host network.
On a different container on the same device, I run PiHole as my sole DNS. Supposedly this does not matter, but I am mentioning it anyway.
The Raspberry is connected via Ethernet cable. I have a Netgear Switch to manage this network.
Normally all Apple devices are in another network with a D-Link DIR-842V2, which is accessible from the cable network - there is a route defined. To avoid this being a problem, I tried to connect to a different wifi, which is basically part of the cable network (created by a FritzBox in this network). This did not change anything.

I tried installing HomeKit bridge via UI and via YAML, using a minimal yaml example:

homekit:
  name: "Testbridge"
  ip_address: "raspberry_pi_local_IP"
  advertise_ip: "raspberry_pi_local_IP"
  filter:
    include_entities:
      - demo.demo

Both ways, it does not work. I scan the HomeKit Pairing QR code with my phone (Home App) and it does not find the Bridge.

Thank you for taking the time to answer. Looking forward to any suggestions.

Sascha