What if we didn't sync the earth and moon rotation, but instead when we run the cable from the earth to the moon, we attach that cable to a little train car that runs on a track all the way around the earth? The moon will pull the car around and it would have generators on it to get the electricity.
I assume the little train car will quickly become a wrecking ball, leaving a band around the earth where you can't build anything and also if you go there you might randomly get absolutely FUCKED by a swinging train car.