OK so I just read the stutter edit manual and I think it should be doable.
Stutter edit works by sidechaining the audio. So I imagine you'd want to hear the audio through Stutter Edit all the time, no matter if you are manipulating it or not. So I would keep the output of the stutter edit track to stereo output, but set up an aux send [i]post fader[/i]. set it to unity by option clicking the send.
Then make the lil' unmute/mute machine in the environment, and hook it up to the aux track. So when you trigger stutter edit by holding down a key, it unmutes the aux and lets the signal pass through.
Create a new audio track, record armed and ready to go. Use the aux channel as the input.
A couple of things I reckon will help:
1. set the output of the original audio track to no output. That way you'll only be listening to the stutter edit track.
2. Also set the output of the new recording audio track to no output, so there is no double up in volume because of the split signal. You'll still be able to hear what stutter edit is doing because the stutter edit track is still set to stereo output.
I don't have stutter edit to be able to try this out, but I'd be really interested to hear if it works!