A way to store all the energy is to use the rectified pulses to drive an inductor, not a capacitor. The stored inductor energy can then be used to charge a capacitor, a sort of simple switching boost converter.
You will need a couple of switches to switch between the inductor and capacitor. You ground the output of the inductor with a switch while the inductor is storing the energy. The you open that switch and close another to connect the inductor output to the capacitor which converts the inductive current into a voltage on the capacitor.