Well, I have to think that after a few cycles, that water is going to have a fair amount of grease/scum floating around on it. Not sure what that would do to the taste of subsequent meals prepared/served with stuff cleaned that way, but it would at least be safe. I suppose you could boil off most of the water at the end and have a lot less (maybe none) left to have to contend with.
With no water to cycle, I reckon the dutch oven method to be superior given the restrictions. Chemical treatment or boiling the washwater might disinfect it, but it won't do much to purify it. Now if you had a microscreen and a good reverse osmosis system, you could drink that water, no worries, when you were done processing it.
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The ultimate result of shielding men from the effects of folly is to fill the world with fools.
-- Herbert Spencer, English Philosopher (1820-1903)