Can't power controller and LEDs with chosen power supply (works in one project but not in other)
Some ideas:

1) Remove the LED strip and continue to test the 5V/4A power supply with your ESP8266 board. Make a single LED flash for testing. Does the ESP8266 board work ok? Power cycle and wait between cycling and confirm the ESP8266 works every single time.

2) Then attach the LED strip. Does the ESP8266 continue to work during each power cycle?

3) Suspecting that you have in-rush current taking place here and is causing the 5V/4A power supply to dip. Could be the quality of the power supply although you are noting that standard USB port is able to power your LED strip? That is a very high current draw for your LED strip if you are selecting a high intensity for your LEDs. The very first thing I would do in the ESP8266 code that is driving your LED strip is to turn on a single RGB LED or a few but only at a visible but dim level.This is to reduce the start up current draw from your power supply. Does that work for you?