I can highly recommend a Kite Circuit Sword Gameboy build like my own.
It's basically a RPi3+, so has a lot more raw power than the Zero builds like RetroFlag, but still no need for active cooling, fits into an original Gameboy case, and there's lots of batteries to choose from. Mine runs a 5200 mAh, which is charged via USB-C and holds power for around 10 hours.
Costs a bit, and takes a bit of tweaking of the case, but everything is pretty much plug'n'play, apart from soldering the speaker and a mode-button onto the board, but really worth it!
I just completed Links Awakening DX on this machine, before playing the new game on Switch, really enjoyed it.