It would appear you have some electrical restriction (probably dirty connections) that is allowing just enough current to pass to fire the ignition, but once you add the extra demand of the lights, you are experiencing a voltage drop to the ignition. May be a bad key switch, or could be any of the low amperage wire connections in the key circuit.
I'd start cleaning connections before I spent any money and it will probably go away. You mention no cranking or charging problems so we will assume they are ok.