It sounds like what you really should be doing is purchasing a 500 dollar game thats a total disaster cosmetically but more or less functional. Then you have your mechanical engineering project in front of you and you electrical engineering project as well. You can do with it what you will but absolutely no one would argue you could not take those parts and make a new game- be it cosmetics, programming, engineering, etc.
Forget about building this from scratch- there is zero chance you have the background to do this- you get that by working in the area for a while. I have a lot of the skills and some of the experience I would need to do this and I have no illusions- building a game would be incredibly difficult. Set yourself up to succeed.