Running electrical for bookcase LED lighting (choose a scenario)
Use low voltage as your secret weapon

LED strip lights are more readily available in low voltage anyway, and low voltage not only makes it a lot safer, but sidesteps most of the Electrical Code.  Now you don't need to think so hard about how to get power from A to B using Class 1 wiring methods! 

At that point I would probably have a DC power supply plugged into the "orange" receptacle, then come straight over laterally to the bookcase, then conceal the remaining wiring in the bookcase.  As far as getting across to the other bookcase, you could tuck a small cable in the lip above the brick.   You have a lot of freedom in low voltage.