One method is digital. Measure the voltage drop across a small resistor and put it to a data logger. Also log the battery voltage. Do math on the numbers.
Another approach is to make a precision current sink (drain, load) so you know what the current waveform is and log the battery voltage until it gets below the threshold. Hand calculate the total charge from the know waveform and total time.
Make a power supply for your device with a gain of one op amp circuit with a low power FET on the output of the op amp and inside the feedback loop. Put the other op amp input to your battery. Use op amps with very low input bias current. This will power your circuit from the exact voltage that the battery has.
Then measure the current flow at the drain of the FET. Put a current drawing circuit across your battery that exactly equals the current at the drain of the FET.
This method includes second order effects like the battery voltage dipping during current pulses.
Monitor the battery voltage for the minimum value you specify or monitor the circuit for proper functioning.
What about taking an integrating ADC. The delta sigma principle AD Converters have as a major part an integrator. So the maximum time resolution must not be as high as the 20us pulses you want to measure. The integrator will do Idt for a short period which equals the charge your circuit took out of the battery during the integrating interval.
The only thing you have to do is adding the charge amounts at a low data rate.