Your probably need to scrap the back of that cart mate, if you want it to look like new.
I hate to say it, but if that cart is indeed melted in the center, then you'll only ever be able to use 2 screws, unless of course it hasn't done too much damage to the front, then you might be able to taper/thread whats left, just enough to get the third screw to just catch. It won't do much to hold the cart together, but its aesthetically pleasing to see the three screws. The hinges and two other screws will keep the game together.
If the cart is held together good and tight, you might need to drill out that hole mate.
To be honest man, if you are serious about getting the cart looking like new, get a donor cartridge first. Take the back and screws of the donor cart and use them for the Kid Icarus cart.
If your really not up for it mate, I say there's someone around here who would be willing to sort that out for you. I'd do it for you, but that would cost you a ton in shipping, as I live in Ireland.