From what you described, I would like the idea of the washer draining into the black tank, and it will keep it really clean and rinsed out well.
But as you also described, you must open the black tank when washing.
Your grey tank should hold at least 1 day's worth of water. I used to live in my RV with a 42 gallon grey tank, and would put away my sewer tank hose on freezing nights, only pull it out once a week to drain the grey tank, while at the same time I would refill my fresh water tank.
Check
CampingWorld.com and check for sewer products, and then look for a tee that will allow your main sewer hose and a garden hose into the same fitting. Then you can put a cap on your grey only line, and run a garden hose to the sewer line. A washer fill line (with two female fittings on it) and a 25' long green garden hose (so you don't accidentally use it to fill the fresh water tank once it is dirty) will complete your set up. Then you could drain the grey tank with the garden hose into the already in place black tank drain hose fitting.
Personally, I would leave the black tank valve closed until after you are ready to start the washer, and then only leave it open as long as you are doing laundry. I would also put a sign on the washer - open tank valve. Another on the dryer, close tank valve (when dryer is done).
Yes leaving the black tank open all the time will lead to only the water going down the drain, the remaining becoming solids inside the tank that will be really difficult to remove. Think of it as 'Poop Cracker' on the bottom of the tank, that can dry out if you let it.
Have fun camping!
Fred.
