This is one reason I keep a five gallon bucket around. If the grey water tank is nearing full and I'm boondocking, I'll fill it up and dump the bucket down the toilet into the black water tank.
Are you sure you simply cannot open the valve of the gray tank, then open the valve of the black tank, and let the gray water flood into the black tank? Of coarse this must be done with the drain cap installed.
You should hear a rush of water. As soon as it gets near quiet, close the black tank valve before it back washes back into the gray.
That would be much more friendly than running around with buckets of dirty water for the same results.
Since I use a macerator pump, that is actually part of my dumping operation when not at FHU (because it is a "free" high volume water flush of the black tank.) However, I don't want to have any water whatsoever leak to the ground, so I try to avoid the drain cap method. It isn't that much water that would come out, but with people watching, sometimes any water dribbling out will set off the pitchfork and torch instinct.