Just a thought... Doesn’t the sleep timer have an amount of time it sleeps in low power,
then wakes up, then you have to put it to sleep again and keep working that way?
If so, you could turn on the output pin and just check the one button for power on before putting it to sleep each time.
That just means you have to hold the power button to turn it on, which isn’t out of the ordinary.

That’s if an interrupt doesn’t work in the first place.