I checked the voltage to the LEDs (gray wire from right handlebar switch) using an oscilloscope and it runs about 14 volts with a short spike (about 1 more volt, I think--did not write it down) at each sparkplug firing. On
http://reuk.co.uk, I found a circuit for a voltage regulator using an LM2940T-12 and a 100uF capacitor. I assembled the regulator on a test board today and it completely smooths the voltage at 12 volts.
Since the LEDs are pretty cheap, they likely have only a resistor to limit the current and so depend upon steady supply voltage. Two to three extra volts will let them draw too much current, which will wreck them. I will see if the regulator helps as I will replace all the LEDs.
I will install it next week after I receive some Hitachi connectors to let me add it without cutting wires.