Bluetooth Button (Remote) support

Hi,
With a combination of bluetoothctl pair, trust and connect, I got to ¿successfully? connect a Hama BRS3, which is just a branded version of a normal bluetooth remote/shutter (which are physically similar to the iTAG devices, but work different. I also have 2 iTAGs in the drawer though.

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However, no event is passed through to Home Assistant (running latest HA OS on a x86 mini PC), even when adding this to configuration.yaml as per this tutorial

keyboard_remote:
  device_name: 'hama BRS3'
  type: 'key_down'

Note: using device_name as per the tutorial, since it’s not generating any /dev/input/eventX

The device seems properly connected, and it connects/disconnects properly when turning it on & off (as in bluetoothctl devices Connected will show it or not as expected).

Any idea what to do from here?
I believe I’m much closer than yesterday (device didn’t really connect, blinked a lot, and was getting
Failed to pair: org.bluez.Error.AuthenticationFailed
when attempting to pair). Luckily I moved past that as mentioned at the beggining.

Thank you