DS3231 Internal Temperature Sensor Accuracy

I would like to add to this post that as a proof experiment, I had placed a DS3231 module in the home freezer for several hours (without the Li cell). On connection to a Nano, it read -9.0 deg C... (the freezer is set at -18C) and within 250 milliseconds it jumped to 4.5 deg C, so I would not trust it for low temperatures.

This was using the ds3231-master library.

BMP280 Temperature: 20.27
01/01/1900 00:01:05
Clock temp: -9.00
21.00 lux
Analog: 2145.00
BMP280 Temperature: 20.28
01/01/1900 00:01:05
Clock temp: 4.50
21.00 lux
Analog: 2140.00
BMP280 Temperature: 20.27
01/01/1900 00:01:06
Clock temp: 4.50
21.00 lux
Analog: 2155.00
BMP280 Temperature: 20.27

The reason I am interested in low temperature performance is for a solar photo-voltaic to lead-acid charge system, where the charge bank batteries are stored in a small shed in a field.

I would not trust it for low temperatures

Why not? All you've proved is that it responds quickly to changes in temperature.
I have had a DS3231 and a DS18B20 monitoring temperatures for 3 years and they always agree within a fraction of a degree down to at least -18C.

Pete

Dallas have a pretty good track record with these devices, but best see getting a temperature from a RTC as a bonus. The DS3231 is a good clock and the DS18B20 is a good temperature sensor, but I wouldn't use the DS18B20 as a clock.