That's an interesting question. In your example the battery is being used as a buffer, it isn't really doing anything since the consumption of your device is less than the available solar power.
You would presumably not want your device to flicker on and off as clouds passed by or birds flew over the panels so we might reasonably expect the battery to do some work.
The installer in me says it would work great with 200 Amp hours of batteries and everything less would have to be by experiment to see what works. You may be able to get away with half that, I wouldn't reasonably expect to go less.