Unfortuneately, the light control box is that way by design, as it's supposed to work like this:
Key on, engine off: Lights OFF (helps to allow all available power from the battery to go to the starter motor during starting)
Running engine idle/low RPM: Lights DIM ON
Running engine higher RPM: Lights FULL ON
I had a warranty exchange with Kymco because mine switches on/off occassionally even when cruising along at 20mph or so. The new box performed exactly like the old one, and Kymco's headquarters just responded, "yeah, that's normal for this bike".
So, I decided it would be quicker if I just installed relays, while swapping out the other standard light bulbs (tail/brake/turn signal/dash) for LED lights to keep the total electrical load on the charging system low (as the headlights now stay on full-bright even during idle). Automotive relays are usually <$10 each (you'd need 2), and are readily available from most auto parts stores or online:
http://www.amazon.com/gp/product/B000P61E36/ref=ox_sc_act_title_1?ie=UTF8&psc=1&smid=ATVPDKIKX0DERAs for the unrestricting the CDI, I don't have any experience with that, so unfortuneately I can't be of much help on that one.