I was basicly looking for the same thing, record just the audio from the system preferably using 2 sound sources. That's not even that much of a problem if you don't use a hotkey for recording the second stream.
You can sort of still use bandicam for this although you'd need some other piece of software which can read audio streams from a video.
What i tried now is setup screen recording to record at 128x128@1fps with a quality of 10 and the audio set to 44.1K 192Kbps stereo. You still have the great bandicam option to only record from your second sound source with the hotkey and you're basicly (mostly) just recording audio. The video data with this setup is <1% of the stream. Audacity can't open the recorded file out of the box but Goldwave can and there are probably other utils that can do the same. Demuxing is also an option, there's plenty of options to open/get the audio from the recording after saving.
(Goldwave might depend on system codecs so you may need the proper codec installed)