Battery Capacity

The battery in question would limit current to the buzzer, per the battery's internal resistance. Thus, the buzzer "demanding" more current is not the constraint, it would only get as much current as the battery is able to supply.

Typical piezo behavior with limited current available (and I just tried this to check) is that the volume of the buzzer starts dropping sharply once the available current reduces below around 30-40% of the "rated" value, but it remains audible (faintly) until well under 5% of rated current.

Also, a piezo buzzer's current demand consists of sharp pulses, much like a capacitor being charged and discharged by a square wave. Thus, the maximum pulse current rating of the battery (10 mA) will come into play, rather than the continuous current rating.

In short:

The buzzer will work fine, it just may not be as loud as it can be. Given that even a tiny piezo buzzer (mine has a minuscule 9 mm diameter) makes an awfully loud sound, this may actually be a good thing.

As a rule, higher current you draw, lower the capacity is. That is because when the call get discharged, the internal resistance becomes higher and the voltage drop as well.

Anyway, in general, the cell developers always try to increase this effect and to lower the internal impedance. But sometimes it is not easy task, together with the other parameters of the cell, for example the shelf life.

That is why there are many types of cells and you should choose the type you need for your application. The corresponding data sheet must contain a data about maximal allowed current and how the capacity is affected by the current.

Mentioned CR2032 is designed to provide small currents for low power electronic devices for very long time with very low selfdischarge. It means that probably you should not use it with very high currents. 20..30mA is the maximum on my estimation.