After a few weeks of testing, I can confirm that my solar charger circuit is working and keeping the esp8266 and epaper display alive despite minimal power-saving techniques.

For demonstration purposes, I programmed the board to fetch the weather forecast, update the epaper screen, then sleep for an hour.

I pretty much just cribbed the reference design for BQ25504, and I should probably learn how it sets the reference voltage for the battery as currently it doesn't charge past 3.8V, but that's not a big deal for my application.

Epaper!

I hooked the prototype up to an xbox chat pad for testing too...