Ideas / Alternate solutions to trigger a switch
You could always have the output for LED go through an RC delay and an XOR gate.
You can then use this output signal and feed it directly to where the button is fitted as it would simulate a button press of 'x' amount of time. 







simulate this circuit &ndash; Schematic created using CircuitLab

I've also done a simulation to show you what the effects would be, for a longer/shorter pulse duration you would need to play around with the value of the resistor and capacitor but only there is lots of help for RC calculations, google "RC Calculator"

Simulation Link