The concept of neon is that it requires AC current. That is why you have an inverter. Sounds to me like somewhere your AC source is not properly shielded, and it is bleeding into your radio circuitry.

I would consider running a dedicated return for it, if reccomended. Going forward, I would just tell people to use LEDs. Get the new ultra-intensity ones, they'll never break, run off of 12v DC very easily, and draw much less power...

-PHiZ