Hi and welcome.

Biological clothes washing liquid, the kind with enzymes that works at low temperatures, is wonderful stuff for getting any organic muck off.
I would try diluting some in room temperature water, and applying it to your duck with a small toothbrush or paint brush on the areas you want to clean, while trying to avoid getting the base too wet.
It might be a bit awkward, you may need gloves to protect your skin, but it should be manageable.
If any of the marks are made by rust, some minced raw onion will (slowly) dissolve it all.