I wanted this to be battery powered, which raises two questions: which cell to use, and in what configuration? Looking around online, I found this chart:

From this it was pretty clear that I should use AA cells. 9V cells only have 20% the ampacity at a much lower current. AAA cells have a similar problem, and aren't as common. The other ones were too unusual.

The battery configuration can be determined by looking at the discharge curve. This chart was fairly typical of what I found:

So the usable voltage range of a AA cell is [1.5, 1.0] V. Four cells in series would be [6, 4] V. If I set my system voltage at 3.3 V that gives me 700 mV of drop out for the regulators, which isn’t hard to meet. A system voltage of 3.3 V was chosen since it’s one of the standard ones and 5V would have required a fifth battery, which would have made the PCB larger. 

The bread board prototype showed that 3.3V produced a sufficiently loud tone.

So a AA 4s1p battery configuration was chosen.