I assume you are using the barrel connector. Could you use the USB connector to provide power? I find a different result on my BBB if I use the USB vs the barrel.
or perhaps I don’t fully understand your issues ???
Well in your application you would need another small CPU monitoring
power and issuing a reset upon power good
there is a battery cape that does just that. I have a design that does
that for my design using a BBB. http://andicelabs.com/beaglebone-powercape/
something i noticed in testing for my design. once things are off there
is nothing to tell the BBB to restart.