I use Calgon in gray tanks AND the black tank. I prefer the liquid.
I do NOT use Dawn in the black tank, IF I use any soap, it is liquid laundry detergent, a capful or maybe capful and half. I think I would stay away from the Dawn. Not that it is right or wrong to use it, I don't think if you used a small amount with a small amount of water, say, 3 to 4 gallons there would be too many bubbles!