Thanks -- to my horror, I discovered that the capacitor was actually 120p. I replaced it with a 100n MLCC and checked the power supply:

Then I replaced the ground lead on the scope probe with a smaller spring thing, and things looked a bit better:

I added a 2.2u electrolytic (pretty much all I had in the 1u range), minimal effect. I also twisted the power supply leads, no effect.
Then I checked the output again. The positive edge is much better, but still exceeds 12.3v.

The negative edge has not improved.

Now, the scope I'm using is a 50 MHz scope with 1 GSa/sec. I'm not sure if that means the analog bandwidth is 1 GHz.
As an experiment, I set up the function generator to output 12V 500kHz and measured that directly with the scope. The rise time shows as 6.4 ns. This is certainly slower than the rise time of the chip output (which appears to be 4.8 ns).
Next, I added a 1.2k resistor in series from the output to the scope. This gave a slower rise/fall time, and eliminated the ringing as expected.


A rise time of 50 nsec does limit the usability to about 20 MHz, which I am willing to live with. In the actual circuit, I'll be using lower resistance -- 120 ohms to provide the maximum 100mA to the switch IC -- which may help things. I tried a 470 ohm resistor also, which knocked the rise time down to 25 ns and the fall time to 10 ns but with a short bounce to 2v before settling down at 20 ns.
Overall, I'm pretty happy with this. However, I am now not certain if I will need Schottkys (
these, for example) on the output. They will certainly add capacitance and thus transition time.