From Taylor's Reddit post (as I couldn't find anywhere else this was listed):
"10-40mA current draw (under normal use)"
This was a year ago from his AMA, so specs might have changed, but lets assume this is still roughly accurate, and let's also assume on the high end/worst-case battery life.
Using what is the correct math as far as I can tell, if we have a 1000mAh battery, then at a draw of 40mA it will last 25 hours. (1000mAh/40mA=25h). Now, what is the mAh of your button cell battery?
The Wikipedia page for batteries lists 'Typical Capacity' anywhere from 25-1000mAh.
The CR3032 is rated at 500mAh and is 30x3.2mm, and is 3 volts.
The CR2427 is rated at 1000mAh and is 24.5x7.7mm, also 3 volts.
These would last about 12.5 or 25 hours respectively, assuming that the Flutter would be okay receiving 3v instead of 3.3v (which I think it would be, but not sure).
Depending on what sort of battery life you're looking for, those appear to be decent options.