I met someone once who had a similar level of shy bowel re: neighbours. Sounds like a nightmare.
Am guessing you've tried a whole bunch of things, so forgive me if this is teaching my grandma to suck eggs, but...
My own hearing has become super sensitive of late, and that coupled with some difficult neighbours and screaming sirens led me to investigate ear plugs. The best ones I found were soft silicone ones from the local pharmacy. They don't go in the ear canal but sit in the little divot of the ear (like a plug over a sink hole) and don't irritate the ear so much. The effect has been to lower the amount of stimulation my ears are picking up and I wonder if you were wearing them, whether they might help decrease the hyper-arousal in your nervous system regarding neighbours being close by. ...Tricking your nervous system into relaxing enough to have a good crap. If your nervous system can't hear them, then it might start to think that they can't hear it, and it might feel a bit safer to go.
Just a thought.