I believe that there are several problems happening here:
- The operating voltage of your LEDs is very close to 12 volts. Blue/white LEDs like to be powered from 3.6-4.2 volts, but if you drop below that on a per-LED basis, they get very dim, very quickly: There's a very sharp voltage "knee" below which they go from being "normal" in brightness (say, 12 volts) to being dim/off (11 volts) - just to through out some totally arbitrary numbers. If you have an adjustable power supply, you can try this for yourself and see the problem.
- The voltage regulators on motorobikes are often abysmal and "approximate" at best. All they need to do is throw something in the range of 12-16 volts to (more or less) keep the lights on and the battery charged and that voltage goes all over the place with engine RPM. Some seem to be shunt type where others appear to be phase-angle to produce an "average" voltage. The current that you are seeing in the supercaps is probably very likely "spiky" - that is, very nonsinusoidal, even more than one would expect from the clipped top of an AC waveform.
The solution to your problem could one of two ways.
- Use LEDs that operate at full brightness from a lower voltage (say, 9 volts) and drop a bit with a resistor. This will give a bit a leeway in voltage variation with less brightness change. Since you already own these lights and want to use them, you probably won't do this!
- A better way is to use a buck-boost regulator to set the voltage at the desired level. These are electronic devices that will take, say, 10-16 volts but always give you 13 volt (or whatever you set). They can be found on EvilBay IF you look closely: There are boost regulators (go from low-to-high) and buck regulators (high to low) but what you want is *specifically* a buck-boost, which is built differently from the other two! If its specifications don't specifically say "buck-boost" then it's not what you want. (You could kludge and get a boost converter to take it up to 16-20 volts and then a buck converter to bring it back down to 12-13 - but that *is* a kludge!)
Using four Schottky diodes will probably give you another volt or so, but it probably won't help that much.
Best of luck!