Then when you press the button the device boots and quickly turns the LED red. Keep holding and the code runs that posts the URL. When the URL replies the LED goes green and you can let go of the button.
Device is initially set to boot and look for network credentials, so you hold the button. It starts blinking blue. When it finds the credentials it turns solid blue. Now let go of button, and press again. Now it switches to red/green mode since it has a URL.
Of course Amazon thing has not shipped so no one has seen it for real.
Your idea with MOSFET is probably even better. Use button push to get CPU initially going, then it can shut itself off via MOSFET.