I've come across a few posts of people doing similar things with the Raspberry Pi and the RetroPie OS or similar, and there are quite a few write-ups of people following this recipe along with a fully 3D-printed body, or putting it inside and existing Gameboy.

Originally I planned to go for the completely custom 3D printed body because it allowed more flexibility and control over design. I started to order parts including a 5" screen and a bunch of cheap console controllers from eBay to salvage parts from (like the buttons) or use as external controllers for multiplayer games.

Then when I got one of the knock-off Wii Pro controllers in my hand I realised I probably wouldn't be able to design and create something this sleek and ergonomic.. I held the screen to the top of the controller and then I knew exactly what had to be done!