Given modern miniaturization I'd guess it would be feasible to create a functioning Gameboy w/ battery and cartridge slot fitting in the envelope of the LEGO "screen". The controls would be a challenge, but I suppose it could just be controllable via Bluetooth and the on-device controls would just be cosmetic.
I don't think the controls would be a big challenge or need to be wireless, Lego already has made buttons and wires. But it would need a lot of custom parts and would cost a lot more than a non-functional device.
