Theoretically, this could work: Pico to Pi HAT, battery HAT, pin-extender socket, Inky Impression. But this would be very unstable mechanically. And it would constantly feed 5V from the battery to the Pico. So you will have to make sure you use intelligent power-management (sleep-modes) from your program.
I have a similar setup with an Inky Impression 5, but I am using my own Pico to Pi HAT and in addition a power-shim for the battery. You can look at this post for details: Weather display project - #15 by bablokb
(scroll up from that link which only shows the image, a few posts up there are a few links to the adapter-hat and the shim).