ruthhope
Crowing
With my pekin that has a deformed bill and cannot clean itself properly, I have never needed to use dawn dish soap. Dawn will remove natural oils the duckling has already preened onto its feathers. Not bathing for a few days will reduce the amount of preening and stimulation of the preen gland. Just by sitting in water, and splashing around, this duckling will wash out any poop that it has sat in, as my pekin has to do to keep himself reasonably clean [although his standards of cleanliness are not up to mine, and I make him bath more often than he would otherwise choose.]If it stays looking wet, I'd bathe it in Dawn Dishwashing soap in warm water and blow it dry with a hair dryer. Then keep it dry for a few days before letting it swim again, so it can re-waterproof its feathers. Keep the brooder very clean and dry and you should be fine.
The OP made it clear in a subsequent comment that the brown color is a result of food dye on the bedding -- its not feces from sitting in a dirty brooder. Plus the OP says it has improved already with increased bathing in water.
I am firmly against using dawn to bath a duck or duckling.