I'll second the warning that the voltage reading circuit may be draining the battery.
You'll also want to make extra sure that the ChronoDot is actually getting VCC. "Well it works" you might say, but remember that the DS3231 will actually communicate via I2C while on backup battery only. That will drain the battery faster.
There are also some settings that will output the square wave signal while on backup power. None of those settings are enabled in example code.
I have ChronoDots a couple years old, running all that time with no VCC and the backup battery is still good.
Finally, just use the Arduino Time library. Anything in there designed to work with the DS1307 will work with the DS3231. It already handles communication with the RTC, syncs at an interval you define, and time is handled internally as an Jan 1 1970 epoch timestamp. Look here for a project I just did with the Time library: http://www.macetech.com/blog/node/115