Voltage drop on some PCBs, but not all?
I based my design off the TPS61099x datasheet and quite religiously
  followed everything in it (e.g. schematic, recommended parts). I
  thought it would work.


Your religious beliefs don't seem to be very inclusive. You completely ignored the suggested layout. Look at your design, and the recommended layout side-by-side:



Do you see how the recommended pattern on the switcher Vin pin is designed? A solid block of copper. Yours is a ratnest of skinny wires. Parasitic inductance will be everywhere. Look at the shape of ground: same thing, a solid extremely wide block of copper versus your skinny wires. I am sure there are a few other problem with grounds as well. 

Switchers are sensitive to details of layout, all MHz-range switchers are fundamentally sensitive to implementation of "high-current" loops. You can't get decent (advertized) efficiency nor sometimes even basic functionality without reasonable implementation of these loops.

P.S. This is the second question for today where OP completely ignored suggested layout. I am wondering why people ignore good , manufacturer's validated layout advices? There is absolutely no reason to do anthing differently. Now you will pay with re-design, re-spin, and your time.

ADDITION: 


  The batteries can handle up to 2A, which is way more than a Particle
  Electron would possibly draw (max. 800mA)


This might be the final clue to the problem: batteries can do 2000 mA, the load can take up to 800 mA, true, but the whole TPS610095 regulator can deliver 800 mA only under most favorable conditions: Vin=3.3V and PERFECT LAYOUT per Mfg's specifications. If the layout is suboptimal in few places, the switcher won't be able to deliver this kind of peak currents, and half of boards may shut off. 

In summary, one margin (800 mA switcher) gets multiplied by marginal PCB layout, and the result is 75% failure rate. You need to increase your margins in this design.