Extract the shadow by turning it into a b/w version using channel effects, use the license plate as a matte for it and use multiply blending mode.
Would you mind explaining that with a little more detail please? I have the duplicated video in black and white, but when I set the license plate to track matte to link (either Alpha or Luma) with the license plate layer as multiply, the image of the license plate either vanishes entirely or becomes a black license plate shaped spot. If I set the black and white video layer as multiply, it appears to have no effect what so ever. Any further details about which step I'm messing up would be a big help. Thanks in advanced!
It would really help to see the shot with the shadow. Depending on the shot I'd probably add a colored solid above the original footage and then set the blend mode to something like screen or add or multiply so that I could see through the layer. The color of the solid depends on the color of the license plate. Let's say you choose orange. If orange was the color then multiply would probably work pretty well.
Then I'd set the in point of the colored solid to the place where the license plate just starts to receive the shadow and start a simple roto of the shadow crossing the license plate. As soon as the shadow has passed the plate you can set the out point. Now you have a colored layer that crosses the license plate. Turn on the replacement license plate layer, move the colored layer above the replacement plate, then change the color to dark gray and set the blend mode to multiply. Adjust the opacity until the shadow matches the original shadow. If the shadow has soft edges then feather the replacement layer as needed.
I think this would be easier than trying to extract the shadow from the original footage because the shadow is going to look different as it passes over the lighter and darker parts of the plate. Here's what I did in a couple of minutes to show you the process.
Roto of shadow: (turns out blue showed up the best)
Adjusting the color to make the shadow look right:
Final Tweak a little mask feather.
If there is a lot of camera movement or there is a bunch of other movement in the shot then stabilizing the shot to do the roto or using Mocha to do the roto may save you a bunch of time. It all depends on the shot, but the trick is using a dark solid rotoscoped layer as the replacement shadow and only doing the composite on the frames that absolutely need fixing.
Inside the stabilized precomp, your license plate is not moving anymore, such that you can easily replace it with another one.
If you just need to change a few numbers, you can do it with the clone stamp. Then you have only very little regions where you need to fix the shadows.
Probably, for reconstructing the shadow, you need some manual masks + color correction. But in the stabilized precomp it will be much easier to get those masks right.
See these tutorials to get started with stabilized precomps:
(mostly chapter 6 is relevant here)