Electrical – Latching Hall effect sensor not working

Minimal circuit could be one chip like an analogue asic but that would cost many, many thousands of dollars. Minimal could of course mean cheap or small or both. Stretching it a bit, minimal could mean low power.

Putting all this to one side, amplify the hall sensor's output, feed that into a suitably referenced analogue comparator. That will produce a high signal when the magnet is close. Feed that into the clock input of a D type flip flop configured as a divide by two and drive the LED, via a resistor from the output.

Neither is very good for your intended task, but the coin cells are better than the A23. In the datasheet for each, you have the Typical Capacity spec (Capacity in milliamp hours at a given load) as well as some information on Load vs Capacity or Voltage Drain vs Capacity.

In short, the A23 is typically 55 mAh, when drained at 2mA (0.002A) (Maybe 0.2mA, considering the 20kΩ load, faulty datasheet dropping the leading decimal point?). Yet has only 25 mAh at a 15mA load. The CR2025 typically has 163 mAh at nearly the same load as the A23. While the discharge characteristics might not be the same in theory, in practice if everything else is equal, you will get 3 times the capacity.

Not knowing how you will actually power the lights, two 83mA lights is 166mA, so if you power them straight on 100% of the time, you will get considerably less than 1 hour life out of the CR2025s.