This page describes useful modifications to the correct the terrible design problems with your DS3231 module.
Be sure to disable the LIR2032 charge circuitry before using a CR2032 battery.
One way to use the DS3231 module in extremely low power mode is to have it interrupt the ATmega328 at intervals, using the *INT/SQ or the 32K output. To make that work, you must have a pullup resistor from the ATmega Vcc to the interrupt input, and remove the pullup resistors on the DS3231 board, as described in the link above.