I’m still thinking about that, just throwing out ideas. Besides, I kinda forgot I kickstarted the other Pi hat until you sent the adafruit one, so I have something to model around in the meantime! 
I might also just chop up some USB accessories and solder them together into some horrible/genius amalgamation and post it on here later… you never know
Speaking of the battery… there is some room under the speakers in the GameShell. So a custom extended internal module housing plus, as you suggested, a larger battery soldered to the existed module’s board might all fit inside. Food for thought