Anyway, my main point is that the battery voltage will decay exponentially.
No. A battery voltage absolutely don't follow the law of a capacitor voltage. On a battery, the voltage drop no so much as long as the chemical process is still producing charges. Depending on the chemistry of the battery, some components might change over time, over temperature, over chemical contamination, or current flow, and this result on a drop of voltage from the nominal one. Anodes stability in for example a critical component in many battery chemistry.
Take time to read datasheet of some batteries and play attention to there voltage discharge curve.