Hey Chris,
This project concept has been tackled by many a maker. We even spun up a version of the popular PiGRRL Gameboy Console (Runs on a RPi Zero, with a battery, controls, screen, etc). Take a look at the writeup.
If you need parts printed, we can help with our 3D Printing services
I hoping to make 3 different versions of a game console.
One pigrrl/psp based., another arcade style console and one is moding a discarded car dvd display to be a game console.