I probably would start with products designed for automotive finishes.
There are cleaners (solvent based), polishers (very fine polishing compound), rubbing compound (more aggressive than the polishing version).
I would start out by using car wash soap and a sponge to get off as much as possible. Then move up the ladder to clear, polishing, and finally rubbing compounds.
I have also found that vegetable oil, if left on the surface for 12 hours or more, will release surface contaminants with little possibility of damaging the finish. On horizontal surfaces I would brush on the oil. On vertical surfaces I would slather on some mayonnaise (about 90% oil) and leave it.
In all cases, I would test first in an area that is not highly visible.
After using any water based products, apply car wax or WD40 to the exposed iron. Cast iron, if left entirely clean, will start to rust in just hours.
I know this, because if I don’t use my car for a few hours, I can see rust starting to form on the brake rotors. If I don’t use it for a few days, the entire rotor surface will be rust covered.
Of course, if it were me, I would embrace the overspray and relabel it “patina”.