Cut two squares of craft foam approximately 4" across.  Use superglue to attach the Circuit Playground in the center of one square.

Add superglue around 3 sides of the other square and glue it to the back of the first square to create a pocket for the battery.

Plug your battery extension cable / switch into the Circuit Playground and your battery into the other end.  Make sure everything's working, and then slip the battery and any extra wire inside the craft foam pocket.

Turn the wig inside out.  Lay the neopixel strip face down (so the lights are pointing toward the outside of the wig) and sew it in place through the silicone sleeve.  Sew the craft foam in place over the back of the strip.  Sew the switch in place along one of the side seams of the wig.

Clip small holes in the fabric of the wig and slip the individual light pipe strands through so they end up on the outside of the wig when it's right-side out.