I've spent the last couple hours looking for the answer to this problem and haven't had much luck.
I've tried a couple different bluetooth headsets with my Moto X (completely stock), and I can get the multipurpose button to launch Touchless Control without issue. Unfortunately it seems to keep wanting to use the X's microphone rather than the headsets. Voice Search will use the headset mic correctly, so it seems to be something with Touchless Control specifically.
Is this even possible? Am I missing an obvious setting?