Here's a Raspberry Pi Zero case that looks like a Game Boy and lets you play retro games

To answer my own question after some research: Eneloop batteries are recommended here and here

Rechargeable batteries are also mentioned in this instruction for n00bs for how to set it up with recalbox, a special operating system for retro games that autodetects the Retroflag GPi Case and already comes with all the screen drivers and shutdown scripts. Nice.

Here’s’ a mod that puts a Li-Ion Battery and a charger inside the Retroflag GPI case, and here’s a mod for a NiMh charging to charge rechargeable AA batteries inside the device.

And here’s the orginal website for the GPI Case, where you can see how it’s made and how it’s assembled.

After researching all that, I will now buy that case, make a total mess searching my house for a spare ZeroPi, and then discover yet again that I find most retro games terribly boring. Ah, well, the kids are going to love it.