How to model/get a capacitor to discharge slowly at first?
An inductor limits the rate of change of current though it (by an amount depending on the voltage across the inductor and the reciprocal of its inductance).  So an inductor in series with the capacitor, and perhaps in parallel with a diode if you want to control the discharge only (and not the charge up), will slow down the early discharge rate of a capacitor.  A bigger inductor will slow down the rate more.