I think you are overthinking this. Some sort of random flashing
effect on the light board, a sound
cue of the same length, and a little practice pressing go on both devices at the same time. As long as there are a lot of flashes going on the audience won't be able to differentiate which flash is which click sound. A little visual chaos and nobody will notice the difference. If you are really picky you could have a start
cue and a stop
cue so you are sure both effects stop running at the same time. But I don't think people will notice if things are not perfectly synced, especially if they are getting blinded by the lights.
I always like to say, if someone notices that this a small tech detail is wrong, there is something very wrong with the rest of the show.