Actually the MPIII can produce power without the regen break if you go fast enough. So it's normal for the battery indicator to wake from death when going fast without battery connected. If I spin the wheel on my bike the CycleAnalyst wakes up without the battery connected.
When you say it is cutting out, do you mean that it goes completely dead or is it more of a brake, go, break, go scenario? If it is the last one I'm guessing you have a bad hall sensor connection somewhere. I had a similar problem soon after getting my SmartPie and it turned out to be a loose hall sensor connector. The female connector wasn't secured to the circuit board, but some hot glue fixed that.
To open the side cover you can't be afraid to scratch the paint because that cover sits very hard and you need to use something sharp to pry it open. Open it on the non controller side if you need access to the halls. I have a video on youtube showing me opening a smartpie, but on that video the cover was already loosened from the sticky water sealant. The MPIII is very similar to open (I have opened one of those too, but I didn't take any pictures/video).