Using 1 light conductor with controlled leakage at each fret marker might be doable but without special equipment it would be hard... Each leakage of light point changes the light level for all the rest of the light escape points
I have experience with fiber optics but I have not done this so consider this theory.

And excuse the long post.
This concept uses dangerous chemicals and you could embed small pieces of glass fiber in you. Breathing small glass parts is not ok. Eye protection is a must. Be careful and think before each step...
Use at your own risk.
Take your favorite colored led.
Select a switch (on/off) and resistor (500 to 1K) to drive it.
Grind off the round part of the nose so it is a semi-point source. You can do the grinding with course to 1600 grit paper. Water is helpful, go slowly.
Get the visible fiber optical cable. I would go to eBay and get a microscope fiber optic bundle maybe a foot or more longer than your neck. This would be a bundle to transfer light so the heat of a lamp does not dry out your microscope slide etc. You do not need the more expensive imaging bundles (all the fibers are aligned at both ends). Normally the bundles use 50 um fiber with about a 125 um first buffer and 140 um 2nd buffer. Cut off the bundle at both ends to remove the connectors. Pull out the entire bundle from the plastic or metal shroud.
Decide how many points of light you want - then add a few to allow for breakage. You can cut these out later.
Separate out that many fibers. Pull the fibers to the side as the fibers in the bundles have a rough surface (think low cost) and will not slide easily but will hang on each other.
Take one end of the bundle and apply some super glue to it - one drop. Set overnight. Polish the end of the bundle starting with 600 grit sand paper working to 1600 grit paper. Go slowly and water helps (better yet is polishing fluid for jewelers). The 1600 grit paper is very course in this world...
Place the bundle against the end of the LED but do not glue yet.
Anywhere you want a light point: cut 1 fiber of the fiber optic. This cut is at a 45 degree angle - so that is the short side is toward the neck. You can cut the fiber with an Exacto knife but you will nick the blade on each cut so have a few extra blades ready and move to a new spot on the blade with each cut. Beware of flying/loose glass.
Using MEK (methyl ethyl ketone) (my ACE hardware has it) (warning this stuff is not good for you - it could ignite if you get too much in the air - read all the warnings on the can - and note it will melt most plastics and remove all the oil from your skin in zero time -
so be careful). Touch the end of the fiber into the MEK (in a small glass jar); the MEK will remove the buffer off of the fiber so light will leak out of the sides of the fiber near the end.
Do this for each point of light you want.
Given each fibers is 0.005" thick with buffer (about 2 pieces of think paper) the bundle size should be workable.
Glue the bundle to the end of the led with super glue while the LED is on. Find some stable way to mate the LED against the fiber bundle end and move the mating surfaces around till you get the most uniform light pattern at the ends of the fibers. Apply a small drop of super glue (using a slower setting dead clear glue would be a good idea). Let set overnight to make sure it is really stable.
If you like what you made then install in the bass neck.
The long fiber bundles on eBay are normally cheaper than the more usable short bundles (for their intended application) get a long one but most of the bundles have lots of fibers in them so you can try a few times...