I did some tests with the Fire TV Bluetooth remotes.
I have an older one without power on and loudspeaker buttons and a new one from an actual Fire TV 4K Stick.
Both can be bind via Bluetooth. To bind it open the Bluetooth Icon on Capsule and start scan. Press and hold on the Fire TV remot for 10 sek the Home button (Button with House on it). Now you see the remote in the list of founded devices. Select it and the remote is bind to the Capsula.
Maybe you need to do it twice (remove it and bind again).
With the new one you can control loudness, power on/off device (standby mode), start the autofocus, go back, go home, moving into all four directions and select an item. Also I can use das forward, rewind, Play/Pause buttons. All is working expecting to control the virtual keyboard in the Google Play Store to search for an app. You can move on the keyboard but not select a key.
With the old remote, I can move in all four directions, select an item, go back, go home, fast forward, rewind and Play/Pause.
Autofocus is not possible and due to the remote has not the buttons for it, you can’t power on/off and control loudness. This remote works so in the Google Play Store App.
But with this remote I have the issue that sometimes after the Capsule hang and neither is controllable by the remote nor the touch buttons on top of the capsule. You need to hard reboot the device (press power button as long until it reboots).
Maybe someone could do also tests with other versions of the Fire TV remotes or there any other Bluetooth TV box remotes. Maybe from Apple TV? But here I think it would not be possible to bring the remote into the binding mode…
Otmar