Shining a light down a clear rod is not really new – i was on a course back in '68 where a "mimic diagram" was being used. Under control, bits of the "wiring" would light up. These were actually acrylic sheet edges, the lights being ordinary bulbs shining into another edge backstage. Almost as interesting as the course material. Normally the requirement is that as much of the light as possible comes out of the other end and it gets called a fibre optic cable. These rely on the sides if the rod being as smooth as possible to make the light bounce off the sides until it appears at the other end, possibly a lot of miles away. If you want the sides to light up, abrade them.
White paint will do little more than cut the visible light down – just use the right colour LEDs.
The size you want, you could be looking for surface mount LEDs. The trick is to get as much light in to the rod (or light guide) as possible.