I've looked up the "alternating controller lights" issue and found that this is typically an indicator of low battery. However, when I check the Xbox menu, I see that I still have two bars worth of power left - I expect most electronics to not bother me until one or zero bars.
The battery, charger, controller, and console are all barely a month old and all Microsoft brand. So it's relatively unlikely that I'm dealing with faulty hardware.
What else could be causing this error condition?