Upshot: The local dealer said it was the heater box and that it would be $3,000 to replace. It couldn't just be fixed.
I was going to take it to an independent, non-dealer-affiliated mechanic to get a second opinion, but given that the heater's functionality hasn't been affected, I've decided to live with it for now. I may still get that second opinion at some point.
If the heater ever stops working, of course, or stops working as efficiently as it should, then I will get the fix done. But for now, I'm keeping my wallet in my pocket and keeping the heater control turned down a couple of notches.
I have the same problem on mine 2012 EX. It comes when is very cold outside. Level of the noise depends on the air speed and temperature level setting and goes away as the cabin heats up. I think this is the air blender door vibrates and makes that sound.
My V assembled in Japan in January 2012.