If Bluetooth is working properly, you should be able to record bluetooth microphone audio the same way you’d record audio from any mic connected to the computer, wired or wireless. Just make sure you point the recording software to the correct microphone input. Usually the easiest way to do that is to go to the settings of your desktop environment (GNOME, kde, cinnamon) and select the correct microphone (this is assuming you’re using a desktop environment for a graphical desktop). A voice recorder app or website should allow you to test whether the Bluetooth mic is working properly if you’re unsure.

After that you’ll need something that can record that supports scripting. I can’t help you there, I have no idea if audacity has scripting support, but something probably exists. But anything that can record audio should play nicely with Bluetooth if Bluetooth is working properly on your computer