You can use any D pad to control your car. Your controller has up, down, left, right inputs. Up can occur simulatneously with left or right, and down can occur simultaneously as a left or right turn.
Anyway, the D pad on a video game controller has the same number of potential positions as your regular bit controller.
Using a button for boost would be an interesting add-on.
To do this mod, you would need to find a controller large enough to hold the controller circuit board, and you'd need to do a lot of jumpering with wires to find out which inputs do what functions.
With a soldering iron and an hour or two free, you should be able to do this without too much of a problem. Good luck