It's definitely the car.
After discussing with some co-workers, the conclusion was the same as was mentioned here: Something in the car's AC/DC converter is flaking out and both overcurrenting and chopping out about an eighth of the waveform on the trailing edge of some cycles.
To confirm definitively, I took apart my Siemens charger (found the previously-mentioned dial to turn down max current), put the CTs on the output of the charger (downstream of the contactor), put the voltage clips downstream of the contactor as well, and the result was identical. Mostly clean voltage waveform, chopped current waveform. I cannot see any way it could be something the charger is doing, even something wacky like a short, as that would affect the voltage at that point.
I'd still like to hear it doing it hooked to another charger (we have one at work, and it seems to now be doing it for the first ~3 minutes of a charge--maybe until it warms up?--so I may test), but this is enough to go to Honda service with and complain. I just hope they're willing to put me in touch with an engineer who I can explain it to, since I'm skeptical a regular vehicle tech is going to have any idea what I'm talking about.
Just musing, two possible internal failures we could think of:
Something in the high-frequency AC/DC converter that is putting load on the charger is flaking out, pushing the current to a peak about 20% higher than it's supposed to be, then about an eighth of a waveform off the peak it realizes that it's overcurrented and cuts out, dropping current to zero for a short period, then it picks back up an eighth of a cycle or so later wherever it was supposed to be on the curve.
The other possibility is that it's overcurrenting to make up net power equivalent to what's being lost in the missing chunk of waveform due to some other internal issue, but the fact that the chopped bit is always on the trailing edge of overcurrented peaks seems to be the opposite of what that would indicate.
For those wondering if they're having a similar problem: Put your ear near either your breaker panel, any LED bulb in your house, or probably a UPS or other electronic power supply. If it is making a rattling noise when your car is charging, and only then, you probably have the same failure although it's going to be a bear to diagnose without fancy equipment. If it isn't making any hum other than what it usually does, your car is fine, and any problems you're having are probably just due to the increased load.
Technical aside: I logged harmonics in the second test I did, and for bonus interesting points that time things were rattling for about two minutes, then they stopped audibly rattling but the waveform still looked like garbage for another minute, then it went to normal.
There were no really high-value harmonics going on, but the 3rd, 5th, and 10th order harmonics were higher than normal and spiky while it was noisy, decreased noticeably when the noise stopped, and then decreased much more when the waveform stabilized. The 7th and 9th order harmonics were higher than normal and spiky when it was noisy, increased when the noise stopped but the waveform was still wonky, then decreased when the waveform went back to normal. Nothing else up through 21st order changed much.
Which seems in my not-an-electric-engineer-oversimplified-analysis say that it's some combination of the 3rd, 5th, and 10th order harmonics that are causing every electronic power supply in my house to rattle.
