There may be a simple and 100% solder-free solution for a few of you, if you are like me, and own an old Pocket CHIP and/or purchased an extra battery for the CHIP device. (Sadly, I don’t know of a source for these batteries, but it’s a good way to use one that might be collecting dust.)
I ordered an Adafruit 2500mAh battery (thanks to everyone for the pointers above in the thread!), along with the package of wires and connectors @QuantumKraken linked to. But when it arrived and I decided to see about upgrading my GameShell, when I went rummaging through my stuff I discovered the spare CHIP battery and checked it out. That battery, it turns out is 3000mAh!
Using this conversion page, the labelled 11.1Wh at 3.7v comes out to 3000mAh:
To hook it up to the GameShell, I took one of the female connectors from that package of wires I bought. (Yeah, I now have 30 unneeded wires with JST connectors attached, and the other 29 JST female connectors as spare parts…) That connector fits perfectly on the end of the white GameShell cable coming from the mainboard.
Just to be sure, before I connected the CHIP battery, I tested it with a multimeter, and the red cable (on mine, at least) is indeed “+” while the black cable is “-”. It also looked that way with the labeling on the battery (as shown in the picture), but I didn’t want to risk harming the GameShell if I was wrong.
Even better news, the battery JST connector and the pins coming off the back of the female JST connector that I’d already plugged into the GameShell cable, perfectly fit into each other and connect snugly. Note that there is no plastic forcing the connection to go a certain way, so be sure to know which way you’re connecting it! I made sure the wire that was on “+” for the GameShell battery went to the “+” on the CHIP battery, and the same for “-”, and everything worked fine.
I was tempted to tape up the connector since there’s nothing holding it together except the pins fitting snugly, but the battery is fully covered and there’s nothing but plastic and air around it, so I just left it alone. I’m pretty sure it’s not going to come apart on its own, and there’s nothing for it to short out on, but do whatever works for you and feels safe. 
The CHIP battery is thinner than the plastic case for the Clockwork battery. I eyeballed it and measured it and the gap is 2mm. I had some spare, white foam rubber sheet that was 2mm thick, so I figured it would compress a little if necessary, and I put it up to the Clockwork battery case and cut around it to match the size. I put the battery (with info showing to the back of the GameShell case) in first, then the foam so it sits between the battery and the case for the controller/buttons. The exterior case felt just a tiny bit tighter when I put it together, but it fits fine and the gear things at the top hold it together easily without anything being forced, and the battery is not sliding around inside.
Hopefully a few of you have an unused CHIP battery laying around. I’d done a bit with mine in the past, but it still holds a good charge and even if it’s not a full 3000mAh now, it’s much larger than the original GameShell battery. There’s a slightly noticeable difference in weight too, of course, but the GameShell feels fine when I’m holding it.
Special thanks to @Zeronaut for the command line battery checking suggestion: Battery Check from CLI
Just for convenience, I modified his suggestion and made a script to load it from the GameShell directly. If you want to do the same for this, or any other command line app with output you want to see, add something like this to a standard GameShell loader script:
x-terminal-emulator -e “upower -i /org/freedesktop/UPower/devices/battery_axp20x_battery && read -p ‘Press [Start] to dismiss…’”
Unfortunately, the output of the command scrolls off the top of the screen, but the important stuff is still showing, even with the prompt.