I found this topic by accident and thought I would report on my success. I have been recording with my muse 24/7 for months now, including during sleep. What I did was attach a new power supply.
I snapped open the left earpiece and soldered a long shielded cable to the board where the battery is connected, removing the existing battery connection (the spots labeled BAT+ and BAT-, easy to find). I cut a hole for this cable in the plastic using normal wirecutters, and hotglued the wire into place to decrease wear on the solder connection. The other end of the cable I connected to the battery of an old cell phone (still inside the phone), and it works great!
I can charge the Muse while recording by charging the cellphone, and if I do it off a DC battery rather than an adapter, there is minimal additional signal noise. I sleep this way, with my Muse plugged into a DC battery through the battery of a cell phone.
I lay on my back most of the time, but the muse does work a lot better if you secure it with a hat or a headband or a bandana, or tie the earpieces around the back of your head. These are especially true while walking around, jostling the thing.
This was a quick and dirty solution. There must be some way to streamline this. But it works.
By now my Muse has begun breaking (rubber holding the front leads degrades and breaks), and I will need to find a newer solution soon.