I would have thought it would depend where you sent the audio stream to. If Audacity can't see the Bluetooth stream, it can't record. I find Audacity is pretty finicky over where to get the audio from, but when I plug the USB audio dongle it sees it, in the list of sources. I presume the Bluetooth is built in (to a laptop?).

You might have a setting in the control panel and I would have thought you could have more than one channel carrying the audio but I don't use Bluetooth. I never fail to record (eventually) and listen on the wired headphones.