I spoke to a friend about this today and he said to build a delay circuit, a simple 555 timer should do the trick. Hook this into the door lock / unlock circuit. When the key fob is pressed, the door lock gets power and powers the 555 circuit, this powers a relay that powers the running lights. The 555 circuit will keep the running lights on long enough to see them.
As soon as I get a bit of free time, I am going to try this out.