The 2040w is just a Pico-W glued to the back of the e-ink display. So every example you find for the Pico-W should work. But keep in mind that the bluetooth-stack comes at a cost: the firmware blob that needs to be loaded to memory. And that is also needed to update the display.
The Pico-SDK has BT-examples for C-programmers. Else have a look at this thread: BlueTooth on Pi Pico W - WORKING!