The closest you can come to this is to use the pause function. Whenever you pause and unpause, a new audio file is created. If you're recording multiple tracks (which I'm guessing is your use case?) you can pause-unpause before and after each track and later delete the unused silent files. You can configure a shortcut for pause by checking configure extra buttons, then pressing configure game controller.

Otherwise, my own guess is that it's unlikely that beware will implement the function exactly as you requested it.

I can't download the zip to check the docs since my work blocks it as a trojan. I don't recall reading about keyboard shortcuts but since I never remember to use them I could easily have skipped over that part.

Is there a keyboard shortcut? If so maybe you can map that using a joystick to keyboard mapper?

I already sent a feature request email to the dev but I'm curious if ya'll have thoughts on the do-able-ness of this:

"Would it be possible to make the audio recording in bgb toggle-able with a mapped joypad button? Like, I get something I want to record, hit right shoulder button to start recording, hit right shoulder button to stop recording when I'm done."