Well, I guess the search for an alternative power supply continues. The TPS61221 samples arrived, and they're a little better than the TLV61225, but not much. It still draws over twice as much as the NCP1402 under test conditions.

The NCP1402 circuit had inductors that were 10 times the value of the TPS/TLV recommendations (47 µH instead of 4.7 µH - with 2x22µF output filter caps). I'm almost ready to try putting one of those in just to see what happens (not hoping for much, unfortunately).