I am not sure how much current the Cree Zigbee module consumes. That would determine whether or not it could be run using a battery.
You could obviously use just the Cree Zigbee module all by itself, without the need for an Arduino or ESP8266, which would help to reduce total power usage. However, the original lightbulbs are designed to be powered by 120VAC at all times. I also believe the Cree bulbs are Zigbee repeaters. Usually, battery powered devices are not Zigbee repeaters due to the increased power usage.
You may want to consider using my ST_Anything project which offers a direct LAN connected solution, bypassing the need for Zigbee altogether. It really depends on what your specific requirements are for what would work best.