Just a small nit.
LED stands for light-emitting diode. Diodes allow electrical current to flow in one direction and block current flow in the other direction. When current flows across the internal junction of an LED, it emits light. This is different than a regular light bulb which doesn't have a sense of polarity. Diodes also have a critical current level that you must exceed for current to flow.
So, if you want to test the LED side of the circuit, it's not going to be quite as easy as testing for continuity with a multimeter. You have to connect the test probes in the correct polarity and you have to exceed the current threshold as well. It's unlikely that you'll do that with the small amount of current provided by the multimeter. The way that you need to test the LED string is to connect it to a known good power supply that can provide suitable voltage and current. You can do a regular continuity test for the rest of the switches and wiring harness.
If the problem is in the LED strip on the side of the neck, it's going to be a really big deal to repair, and not something that anybody will want to do outside the factory. The way that the LEDs are placed in the neck is that a slot is routed along the entire side of the neck from headstock to neck joint. A thin filler strip is produced to fit in the gap and LEDs and LED wiring is installed on that strip, then inserted into the side slot. If an LED or the wiring has failed in the strip, then I think they will have to reroute the slot and replace the strip. As you can imagine, this is a very delicate, complicated, and probably expensive process. I don't think that you can remove the existing strip in any non-destructive way.
I had one of the rare graphite-necked Series II basses built back in the 80's. It was originally constructed with insufficient neck relief and tended toward having a very buzzy action (I play with more relief than most, I think). As was the regular practice with Modulus-constructed necks from that era, there is no truss rod.
After many years, I decided to address this problem to make it right. I have a lot of Modulus instruments and was friends with Rich Hoeg, who was the Modulus custom shop builder and master setup guy for many years. He took a look at the instrument and told me that he might be able to correct the problem by grinding the necessary relief into the existing fret tops, but that he was unwilling to pull frets or plane the fingerboard for fear of upsetting the LED strip at the side of the neck. I suspect that this is a oonsideration for any LED Alembic, not just the graphite ones.
David Fung