Yes, I have done what you are planning, although I milled the slot 1/4" wide to match the existing skunk stripe, and capped it with a matching piece of wood. But yes, you could mill a narrower slot down the center of the skunk stripe, and fill it with the TOW and black-tinted epoxy. Mill the slot neatly, and you get a neat black line.
A 6K TOW strand, saturated and lightly twisted, is only about 1/16" diameter. So, yes, four will easily fit into a 1/8" x 1/4" deep slot. Four strands into a 1/8" x 1/8" deep slot would probably fit, but might be a little messy getting them in there. Actually, two 6K strands in a 1/8" x 1/8" slot would be plenty of strength, particularly if it's right at the back surface of the neck. I've never actually measured the tensile breaking strength of 6K TOW, but it's in the thousands of pounds range. Carbon fiber TOW is about 5 times the tensile strength of steel.
My favorite epoxy is West Systems, mostly because I've been using it for 25 years. System 3's T-88 is also an excellent similar type of epoxy. Hardware store epoxies that are long-cure, like 24 hours, are probably okay. Don't use any of the quick-cure (5 minutes, 1 hour) type epoxies for this. They are too thick and rubbery. You want a fairly low-viscosity epoxy that will soak into the TOW and dry hard.
A tip: My favorite pigment for making epoxy black is Smooth-On's So-Strong Black Colorant. A teeny drop makes West Systems epoxy jet black.
Milling the slot in a milling machine isn't hard. Use a normal end mill, normal cutting speeds. The key thing is getting the neck clamped down solidly so it doesn't chatter or jump. You've seen my fixture that I used in the MusicMan truss rod replacement thread? Simple to build and makes things easier.
Replacing The Truss Rod In A Skunk Stripe MusicMan Neck