Hi Onno, greetings from a fellow Alembic Hagenees!
I think there was a problem with the earliest Distillates, which had an on/off LED, where the LED actually drained the battery. If this is the same type of LED, I wouldn't reconnect it.
Does it have two switches, or just one? A Spoiler should have the famous Q switch, which works together with the filter. If there is just one switch, then perhaps the LED was wired to be off when the pickup selector was in the mute position.