electronic switch to control battery from Espruino GPIO

the coin cell shall power the Espruino Pico and the display as well, i am not sure about this 3.3v rail, what do you mean?

Usually, you have 5v coming in from USB... That goes through a voltage regulator which brings it down to 3.3v. The problem is the voltage regulator 'loses' a bit of voltage, so say you have input voltage V, the voltage coming out is either 3.3v, or V-0.3v is V is less than 3.6v.

So if you feed it from 3v, the 3.3v rail only gets 2.7v. To fix that, you can just cut out the voltage regulator and connect the 3v battery straight to the 3.3v rail.

What type of display is it? Could it be that the display is actually working, but the contrast is different because it's running off lower voltage?