Electronic – Shut down circuit sections

The first circuit on the page you linked to is already pretty efficient in that just about all supply current is used to light the LEDs. So the issue isn't lower power as much as to be able to run from a lower voltage.

Typical coin cells, like the common CR2032, put out about 3V. This mean you have to use LEDs that light at somewhat lower voltage. Fortunately common green LEDs typcially light at around 2.1V. Switching to green LEDs allows for using a 3V supply.

To cut down power, you have to cut down the current thru the LEDs. This directly effects brightness. If this device is for indoor use, even 1mA may be good enough to noticeably light the LEDs. Getting decent LEDs with decent efficiency definitely helps here. Let's say you want to draw only 2 mA from the coin cell. The voltage drop accross the resistor in series with the LED will be roughly 800 mV. 800mV / 2mA = 400Ω. That's a starting point. Then you can trade off battery life with brightness from there. Lower resistors will give higher brightness and shorter battery life.

I think you can fix it as follows:

Put a pulldown on sysoff, maybe 100k. Just make sure it is strong enough to overpower the internal 5M pullup. Also keep sysoff connected to your IO expander. It should be configured as an output driven low. But when you want to power down, have the expander drive the output high, which will overpower the 100k pulldown, and turn the system off.

Once the system is off, your rails will all collapse, so the IO expander will be dead, and the 100k pulldown will ensure that sysoff is now low so that startup will be possible the next time charger is attached. You will need to make sure that the IO never goes high during bootup or normal operation. For example, program it to be low before you program it to be an output.