Implementing a safety lockout through ladder logic

Your circuit is not quite correct. If V1 is present and you close the switch V1 and the battery will be shorted together, and you don't want that. Their voltages will never be exactly the same, and that will cause a large short-circuit current. More about this in a minute.
When V1 is present the LED will be on through the battery, regardless of the position of SW1.
If V1 is gone the relay will be off, and so will the LED.

How to solve these issues? Use a switch-over relay to switch between V1 and battery and place it at the node next to the LED's anode.

A minor thing, though it will not change the circuit's functionality: you're interrupting the ground with your relay. Don't. Swap the relay contacts and the LED, so that you interrupt the positive voltage instead. Just a good habit. Has been fixed anyway in the previous paragraph.

And don't forget to place a series resistor with the LED. For a 20mA indicator LED a 1k\$\Omega\$/1W will do.

And clean it up a bit. The ground line at the bottom, going to the contact of SW3 is superfluous, and draw a single ground line for both power sources, so that it's obvious that they're connected.

edit (re your revised circuit)
We're almost there. The switch is in series with the relay coil, so closing it will create a path from coil through R1 and LED to ground. The - of the coil should go directly to ground, and the switch should go between V1 and the unused relay contact.
But schematic-wise it looks already much better, don't you think? It may even pass the olin test ;-).

For sensors, I'd look at passive infrared (PIR) detectors first. Those are triggered by warm objects like human bodies moving around. You probably need a few per classroom to cover different angles and differnt movement planes. A false positive is better than a false negative.

These things produce a analog or digital signal. These signals can be measured easily enough by a microcontroller. The microcontroller collects all the signals, in the case of analog decides whether each one is indicating motion, computes a aggregate yes/no motion value, performs some time lengthening on motion detected, and then produces a single digital output for the lights to be on or off. To turn the lights on and off from a digital output, I'd use a plain old relay. The relay can be driven from the digital output via a transistor. The output of the relay then acts just like the wall switch to turn the lights on or off.

I mentioned only briefly that the micro should time-extend the motion signal to make the lights on/off signal. This is important to keep the lights from turning off frequently when everyone is still for brief periods of time. I'd probably keep the lights on for 3-5 minutes after the last motion was detected.