How do I track the strength of a magnetic field to 1:20,000?
Rotate the magnet at a fixed RPM or move it precisely in linear motion and use a pickup coil.. A similar principle is used in commercial VSM (vibrating sample magnetometer) instruments. 

Of course you can use a DC magnetometer (flux gate or Hall or GMR etc.) but the magnetometer will be temperature sensitive and will respond to the earth's field and whatever ferromagnetic materials around it are doing to the earth's field, which in turn may change with temperature.