Rasp. Pi Pico W does the heavy lifting: it contains the buck-boost converter allowing a wide 1.8V to 5.5V to power everything (2xAA, 3xAA, LiPo, Li-ion, to name a few possible battery possibilities), as well as the CYW43439 WiFi 802.11 2.4GHz + Bluetooth combo chip.

This seems like a very straightforward use of the Pimoroni Badger 2040 W board, as well as a common use-case of E-ink. That is: sleep for extended periods of time, wake up occassionally (maybe every 15 minutes or even once-per-hour). Connect to wireless (bluetooth has less power, but WiFi is simpler), download some data, update the screen, and then take advantage of E-Ink's zero-power consumption as you fall asleep again for 15 minutes to 1-hour (or longer).

The few seconds you're up + using the radio is very little power usage overall. Allowing for extended battery life. Use deep-sleep states and timers to minimize sleep power and maximize practical battery life: if you spend 99.99% asleep, then its all about cutting down on the sleep-state power, and not the 0.01% of the time when you're forced to spend 300mA transmitting radio stuffs.