It seems that the sole purpose of the 328P is to manage power. Are you sure you need that? I thought the ESP8266 has its own power management facility.
Here some an article about DHT11 and Wireless power consumption which might help: https://www.hackster.io/Talk2/temp-and-humidity-sensor-with-a-cr2032-for-over-1-year-580114, specially the part where you can optimize the DHT11 readings.Also, ESP8266 is far from the best solution for low-power because of the very high power consumption of the WiFi module together with the long time it takes to start-up/join the WiFi network.
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