If you can find some old flatbed scanners or those neon tubes they use for back lighting in computer cases they have the inverter/ballast built right in. They take 12 volts and are easy to hookup as well.
However the 9 volt battery concept wont work long enough to be worth it. Neon is not all that efficient power wise. I have a 12 inch scanner tube that takes 4 watts. that would drain most 9 volt battery's below working voltage in a few minutes.
Plus good luck keeping fragile glass tubing in one piece on a bike for more than a block or two!
LED light tubes would be far more practical and able to handle the physical conditions plus would have much higher operating efficiency.