Having just replaced Rev 1 of the amber running light with Rev 3 (about which, more later) on Mary’s Tour Easy, both the front and rear lights began blinking erratically. Given that they have completely independent circuitry, this strongly suggests a power problem.
Herewith, the headlight circuit voltage:
The voltage should be a constant 6 or 6.3 V, depending on which description you most recently read. That is the case with only one light attached, so the problem occurs only when running both lights.
The four pulses come from the amber LED’s Morse code “b” (dah-dit-dit-dit) with a 85 ms dits; the first dah pulse should be three times longer than the dits and definitely isn’t. The rear light’s red LED stays on continuously, except for two dark dits, so it draws a constant current and does not produce any changes in this trace.
Both lights have 2.0 Ω sense resistors setting the LED current to 400 mA, which corresponds to 250 mA each from the Bafang controller’s 6.3 V headlight circuit. The headlight circuit’s total of 500 mA should work fine, although the “spec” seems to be basically whatever the OEM headlight requires.
The Rev 1 amber light ran the LED at 360 mA with a supply current around 450 mA. That light and the rear light on the back ran fine, so the supply seems to have a hard maximum current limit at (a bit less than?) 500 mA.
The least-awful solution seems to be backing off both LED currents to 360 mA to keep the total supply current well under 500 mA.
The Smell of Molten Projects in the Morning 
Second least bad: abandon the headlight output and use a separate buck converter. You can get a 3 amp one for about a dollar.
Or embrace the chaos, lose the arduino and allow lights to hickup instead of blink :)
Except it’d be a stepdown directly from the 48 V battery, so I’d need a relay driven by the headlight circuit to shut it off between rides and someplace to tuck all the additional gadgetry. I might (eventually have to) do that, but I really have some other projects on the to-do list ahead of it.