How can I store energy from short pulses of current?
Storing energy in a capacitor is fine providing that the energy requirements are not that big. Consider that the energy stored is: -

Energy = \$\dfrac{C\cdot V^2}{2}\$ where C is capacitance and V is voltage across terminals of the capacitor.

Note that when storing energy into a capacitor, storing more volts as opposed to storing the same voltage on a bigger capacitor is more effective for energy.

Say you used a 100uF, 2000 volt capacitor (you might have to build one out of smaller values of course) and, charged it to 1000 volts DC. The energy stored is 50 joules and you could probably extract 80% of that energy and dump it into a battery.

Ask your self how much energy you'd like to extract from the braking process.

A better method might be to use a buck regulator that fed the battery and forget about the intermediate capacitor as an option. If you were using a capacitor, you'd need some form of high voltage buck regulator anyway to down size the 1000 volts to a battery.