Can I somehow use a neon bulb as a switch?
You can try this. 27V is 3x 9V battery in series. 

If the LED stays on after the neon triggers you may have to play with the voltage or R1 value. 







simulate this circuit &ndash; Schematic created using CircuitLab

The gas in the neon lamp is ionized by the high voltage which causes current to flow, a bit like an old-school thyratron or xenon flash tube.