What is the best way to approximate average current or count battery amp-hours?
I'd use a low pass filter, to keep the average DC component of your sense signal.

You would have to tune the filter to the level of smoothness you want at the trade of settling time.

You may want to feed both to the micro though so you can measure both average and instantaneous levels.

Benefit of analog is also you do not miss spikes that may be in tune with your sampling time.