It also seems that, if each kilometer is thought of as expanding over a period of time, wouldn't much of the expansion between us and the other star be happening behind the light as it traveled? To my mind, this means that any light we're seeing didn't actually travel the entire distance from there to here.
You need to think of this in terms of quantum mechanics. The probability wave of the photon doesn't collapse until it is observed. Once the photon is observed only then can you make sense of it's path. IOW, it doesn't make sense to talk about space expanding behind the light.
The speed of light is not a law. It is THE law. All observations change in order to keep this law in place. This is why you get redshift. The frequency of the light changes so that the velocity of light and the energy of light is conserved. The way in which you observe the passage of time and distance all change in order keep the observed velocity of light the same.