ESP8266 is a popular low-power WiFi chip, you can use it with a separate micro-controller but for many applications you don't need one. "ESP8266 consumes less than 12uA in sleep mode and less than 1.0mW (DTIM=3) or less than 0.5mW (DTIM=10) to stay connected to the access point." [0] Elsewhere it says it can wake-up and transmit within 2ms but I'd take that with a grain of salt.

I'm having trouble finding the same details about the MKRFOX1200 but for the transmitter "Low current consumption: 32.7mA during telegram transmit [...] Typical OFF mode current: 5nA." [1] The processor part is a SAMD21.

My guess is it's possible to get comparable idle/sleep power draw regardless of transmission method but WiFi will require more power to establish a connection and transmit the first byte.

[0] https://cdn-shop.adafruit.com/datasheets/ESP8266_Specificati...

[1] http://www.atmel.com/Images/Atmel-9372-Smart-RF-ATA8520_Data...