I am kind of thinking along the same lines as cnd--that the input source may be getting switched around when Bluetooth device is connected. Check to make sure your input sources and output devices are correct, and that they are the same on the computer and in Audacity.

I recently was recording from a MIDI device into a sequencing program. I opened another program for some random reason, and when I went back to the sequencer, I couldn't record from the MIDI device. I went through about an hour of USB plugging and unplugging and punching and un-punching sequencer tracks before I realized the sequencer had selected the random program as the MIDI source. I know to check now.