Simply a dead battery, kinda
I pulled the UPS out again last night so I could get at things to investigate the problem. Here’s what I found: The battery voltage dropped to ~2.5V after the Pi started booting. It behaved the same when the Pi was powered directly from the 18650 boost circuit (with the rest of the circuit disconnected). Another 18650 cell was swapped in and the Pi booted fine and ran until I shut it down manually. With the new cell in place and the rest of the circuit connected it all functioned as expected. So I've concluded that the cell had become discharged, as it is permanently powering the boost circuit, and wasn't able to support the Pi's power requirements for boot up. I knew that the on-off switch only disconnects the output to the USB socket, but underestimated the boost circuits drain and the impact it would have when used in this manner. Compounding the problem is the DC-DC converter supplying the Pi and the cell; it can't deliver enough current to both charge t