I have reproduced this issue as you have described, finding that only one input device, the internal microphone, is being reported when a bluetooth headset is connected.
It seems that Bluetooth input is something we have avoided historically due to latency issues, though there have been previous cases of getting bluetooth headsets working by setting the audio manager to MODE_IN_COMMUNICATION. I couldn’t get that working, but it might be a good place to start looking. Otherwise, I have passed this on to the Dev team to investigate supporting bluetooth input devices in the future.