So, I had noticed the the Nintendo 3DS battery has the voltage/amperage also. It's odd, however, that others I have seen had dimensions that wouldn't fit the studio. Perhaps their dimensions were wrong, or perhaps this one is, but here's a nice affordable battery that, if their dimensions are correct, would fit the Studio quite well - you'd just have to solder on some wires and you'd be good to go:
Alright, that's the end of my research on this for now. Perhaps I'll send Jack a note and see what he would ask for a replacement battery.