Device/method for handling rapidly changing current demand in DC source
Is there a device or component I could add that could draw an
  essentially constant amount of current from the rectifier, but handle
  the rapid changes in load to maintain a constant voltage?


It's called a large capacitor


  A (very) large capacitor across the device could do it, but I'd prefer
  something a little more elegant and less likely to kill me (or at
  least advice on how to use something like that safely).


I'm still saying capacitor and to avoid getting harmed you should consider interlocks and/or a discharge relay that removes charge within several seconds when power is removed.

Whether you use a capacitor or not, it's likely that if you don't take care, the 200 volt supply will electrocute you.

You should also consider that if one of the coil's reverse diodes failed, the electric arc produced from 6 mH and 100 amps is about one-tenth of the energy delivered by a fairly standard heart defibrillator. How safe do you feel given that the catch-diodes you are using have a peak repetitive forward current of only 20 amps and a non-repetitive current of only 150 amps. 

I'd be concerned about safety with or without a capacitor.