You are right, finally the MCP1725-3302E/SN seems to be a much better choice with a quiescent current of 120 μA and a maximum current of 500 mA.
Furthermore, it has a shut down pin designed to reduce the quiescent current to less than 0.1 μA which is quite perfect. The Espruino board could control this pin when there is no need to provide power to the ESP8266. In fact, that would be a much more reliable solution than a huge capacitor.
Espruino is a JavaScript interpreter for low-power Microcontrollers. This site is both a support community for Espruino and a place to share what you are working on.
You are right, finally the MCP1725-3302E/SN seems to be a much better choice with a quiescent current of 120 μA and a maximum current of 500 mA.
Furthermore, it has a shut down pin designed to reduce the quiescent current to less than 0.1 μA which is quite perfect. The Espruino board could control this pin when there is no need to provide power to the ESP8266. In fact, that would be a much more reliable solution than a huge capacitor.