I think maybe I've figured out what the issue is with the boost converters.
Both the TPS61221 and TLV61225 are hysteretic topologies. The NCP1402 was a PFM controller.
I've put in two more prototype design boards, one for the TPS61097A (also from TI), and one for the Torex XC9140C331MR-G. It's the latter one that gives me some hope - the datasheet specifically describes it as a PFM controller.
I have only read a little bit of the datasheet, but I am buoyed by the fact that the Torex and On Semi parts both claim to work the same way.
The Q:250 price at DigiKey is, unfortunately, 19¢ more, and it appears as though the supply is a bit on the constrained side. But in any event, the next step is to test it out. Boards and parts should be here soon.