In case anyone was waiting for my results:

Well, I successfully accomplished my goal. I simply used a 12v to 5v Voltage regulator from Radio Shack, and cut and spliced into the voltage lines running into the gps unit itself.

Now even when the computer is off, the GPS unit stays on, and keeps it's Satellite Fix, so that way when I power up the computer again, my position is immediately displayed in MediaCar, rather than having to wait 5+ minutes for a fix to be established.