Back to your actual question for a second. Are you concerned about just the grey water? If you are mostly concerned with the grey water, and if your campground has disposal basins for grey water, you could just get a 5 gallon bucket, attach a garden hose adapter to your dump pipe. Open the grey just a little to control the flow and drain into the bucket. Then lug the bucket to the basin, or drop it in the toilet/black tank. You can do the same sort of thing with a tote, but, I'm thinking this could be a problem with the blank tank as others as mentioned. Well, higher "risk" anyways LOL.
I have a 7 and 9 YO along with the wife and I. No matter how much we try to conserve we will fill up a 32 gallon grey tank in two days almost always if we do not use the campground facilities ( or if there are none). But, normally, we can go a good 5 or maybe even 7 days before the same sized black tank will fill. So, transferring some grey to the black tank via bucket might be an option to think of.