You might want to take a look at this retro handhelds spreadsheet. You can try filtering out good GameCube support (column S, support level A) and going from there. Good luck!

Gamecube is old enough that a fairly mediocre phone running Dolphin should be able to do it.

Might not be the absolute cheapest option, but a steam deck is a pretty great solution

And it’s also a steam deck.

Maybe look for second hand 64GB models near you if you’re on a tight budget, you can expand the storage with an SD card easily

Rg cube. But really a good Bluetooth keypad for your phone like a game sir is going to run better and be cheaper.