You could even convert that extension to a complete "hand rest" for your one-handed handeld (love the name ). Not only will it be more confortable, but it would add a lot of space to integrate the Pi and the battery.
The MintiPi already has a fixed format (the mintbox). It would definitely be worth going for a standard GBZ build and playing around with the case modding and the controller. It will need A LOT of modifications though. Basically instead of the cross button you have a joystick to configure and A/B buttons on the side (like how you picture suggests it).
This is so interesting. Great thread.