This is not scientific, and I know little about filtering/reusing grey water, but I have owned RV's and boats with waste water tanks for 30+ years. In my experience, grey water is just as nasty (and smells much worse!) then black water.
I can not see how the complexity, energy use, cost and most importantly, risk, would make recycling waste water worth the potential problems.
In our RV, we hold 96 gallons of fresh water and our camping is 95%+ dry camping. With 3 people and 3 dogs, we can camp and have plenty of water for 10 days easily, with with a little effort, we have gone 14 days and still had a nice reserve. This includes military showers, cooking, dishes, dog water, etc.
The 3 dogs drink a 1-2 gallons a day, and we use about 1.5 - 2 gallons per person/day. This is not scrimping.
My $2 worth.