Just to follow up, the issue does in fact seem to be the battery. To dive in with a little more detail, the iPod starts up with the Apple logo and spins up the hard drive. At the point where the drive begins to unpark the head, the entire unit shuts off and the hard drive re-parks. All because a running circuit board, hard drive motor and head mechanism all exceed the current output of the battery pack and charger combined. (I would assume even when charging, the iPod uses a little more kick from the battery to unpark and move the arm and heads)

The iPod Video in the parts bin at work is finally back together, but the screen is shattered (including the spare unfortunately), the hard drive is stuffed and the battery won't retain a charge. I threw it back in the box and left it at that.