Can On-Off shim power up as soon as power is supplied?

I’ve got a Pi Zero and an InkyPhat, which interrogates the API of my account at UK Energy supplier Octopus. It then displays my energy usage for the last 8 days as a graph on the Inky. This all works, and I’m happy so far.

BUT this only needs to run once a day to update the display. Since it’s an Inky display, it can be powered off the rest of the time. But how do I achieve getting the Pi to start up once a day, run it’s update then shut down again AND not consume any power when “off”?

Given that the pi zero only needs to run for a few minutes each day, I thought I might power it from a cheap solar garden light with a boost converter to provide 5 volts from 1 x NiCad AA. (The light will charge during daylight, when it’s dark it switches on. I want the pi to boot, update it’s display, then switch itself off again until the next daylight-night cycle)

When the pi shuts down normally, it continues to draw a little current (around 50mA I think). I’d prefer to avoid this so that the AA battery can get and keep maximum charge, especially during winter when there might not be much sun to charge it.

I tried out an on-off shim, and yay - zero measurable current draw after a software shutdown, not even using the the daemon.

But when the power is re-applied, the pi won’t boot until the button on the on-off shim is pressed (as designed).

Permanently shorting the BTN pads does allow the pi to boot as soon as power is supplied, but when it is shutdown it will immediately start up again, as the “on” button is shorted.

My assumption is that I can’t do what I want with the on-off shim, but I wondered if anyone else had a mod or suggestion as to how I might achieve this?