Bunnie discussed this on one of his blogs. The electronics used by the Classic do not support a battery, but the Chumby One electronics do - however, the Chumby One electronics generate significantly more heat, and charging batteries *also* generate a lot of heat.
Unfortunately, the stuffing used in the Classic is a great insulator - which means that it's likely that C1 w/battery would overheat if put in the "classic" enclosure. The Classic dissipates its heat through the display panel - the C1 generates too much heat to do that.
That doesn't meant that it's out of the question, but it would require some careful redesign.