Trouble adding led indicator to circuit

[Big Kev-O]

I have electronic locking differential in my truck and attempted to add an led indicator light but it does not illuminate when the differential coil is powered up.
I do not want to use one of the modern automotive switches that have a built in led indicator. I am using a heavy duty commercial toggle switch that I mounted in the instrument bezel choke hole. I wired the led in parallel with the locker coil. The led will illuminate when the locker is not plugged in but will not come on with the locker connected.

I do not understand why the led doesn't work. How is my circuit any different than those cheap led switches they sell at the auto parts stores? Do I need to add a diode or decrease the resistor size. Could it be the coil has so little resistance the voltage going to the led drops?

Thanks for looking, here is a diagram of the circuit.