It might work if you can keep your grey water just plain soapy water but have you ever looked at what comes out? Even if you wipe dishes and utensils practically clean before washing you are going to get food particles. Then after a flush you still need clean water to seal the toilet bowl. Also, grey water can get pretty foul smelling.
I would not do it because I am pretty closely balanced out with fresh water = grey water + black water tank capacity. So when I dump I usually coordinate and add fresh water about the same time. The other reason is I dump black first and then grey second which helps flush out the hose and system and rarely need a lot of water provided at the dump station to do much after cleaning.
2015 Advanced RV Ocean One Mercedes Benz Sprinter
Previous Class Bs:
2011 Great West Van Legend Sprinter
2005 Pleasure-way Plateau TS Sprinter