I guess I don't really have an idea of the size of the Ice Tube Clock, I had imagined it was pretty big but if the ChronoDot can't fit inside, I guess not.
Even if the ChronoDot can't fit, you could still find a wide-SOIC to DIP adapter board and make your own little DS3231 board. It doesn't really need any external components besides I2C pullups, you could squeeze the battery somewhere and run wires. That way you should be able to fit everything inside the clock.
The DS3231, if you get one running, will be headache free and far more accurate than any software trimming scheme. Any of the software or analog frequency calibration schemes are flawed as soon as you start having temperature variations. With the DS3231, you could put the clock outside or on a sunny windowsill or next to a HVAC vent and it will still keep accurate time. The temperature compensated crystal works great...for me the DS323x is my most favorite Maxim product and one I actually think is worth the money and hassle of dealing with Maxim.
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