DROOOOOOOL you git!
Beautiful car and to be honest you look like you have the money to build a seriously nice system. I would definitely go with a tiny pentium Mobile system. I would suggest athlon mobile using nForce 2 so you can get custom resolutions but as you are using a 4:3 screen you can run standard resolutions so a pentium M board with intel extreme 2 graphics will be fine.
If you have the money go for a single board computer style PC - they are quick and very efficient and small. Otherwise look for a miniITX P4 or pentium-M board (i can usggest a few to you for around £180-250). Don't go for too high spec on the CPU, you don't need it.
My one concern is that screen. How does it connect to a PC? do you have a VGA controller or is it LVDS or a pure TFT (laptop style etc etc)?