After a look around, I'm reminded that Inobu provided a description of the lightboard and some of the issues with using an FL12 with it in
his thread on DCC converting a Kato 500 Series. There wasn't a detailed description, because he didn't use an FL12 (although his final statement seemed to be that if he had to do it again, he would; but I'm not sure I'm reading the postings there correctly).
In that thread CaptO suggested that using a TCS decoder's motor output (to simulate the track voltage and polarity the Kato lightboard was expecting) was a better method than wiring in an FL12. TCS was needed because they let you control the motor with a function key, which isn't typical.
The basic problem is that the Kato lightboard expects full track voltage (it contains a dropping resistor or other circuitry) and polarized power to select which bulb is lit. The FL12 outputs two separate "track voltage" signals of a set polarity (and trying to mix them is just asking for a short circuit). You need to separate the circuits for the two bulbs, while still ensuring there's a dropping resistor in each path, and then wire up so each LED is getting power of the correct polarity when the function is on.
Since the FL12 output polarity isn't documented, you'd need to measure it to find out which pad is + and which is -. You don't really need to figure out which is forward and which is reverse, just put the headlight on one set for one car, and on the other set for the other car, and swap the cars if you get it backwards.
If you can take out your 500's lightboard and take a close-up photo of it, someone here may be able to advise on what to cut and where to solder to connect it to the FL12 (or maybe Inobu knows and can make a suggestion).