I know you're in a hurry to relieve your son's suffering GS but it seems like you're jumping all over the place with treatments like the more the merrier.
With this kind of thing where there is a definite, acute cause the best thing to do is clear the system of everything questionable and feed only what you are sure isn't harmful.
So that means the cessation of
cooked foods
Sugars (even fruits!)
Exposure to any kinds of chemicals
Any foods with known reactions.
Meaning that you son should be eating only a very narrow group of known-safe nutrient dense foods. Typically that would be something like beef organs, muscle, fat and marrow. And everything else should be cut out. Raw green veggie juices are ok too as long as they are not root or high sugar veggies. The only other thing I would agree with is the green pastures fermented cod and/or skate liver oil, it is different from all other supplements in that it is essentially a food, not a fractured, highly processed substance. It is essentially high cod liver or high skate liver.
Then you wait for things to clear up.
Once they do, you expose him to extremely small doses of the most likely culprit, whether it be the duck or amway chemical or whatever you suspect most. Then you wait to see if there is any reaction, don't just look for skin reactions, notice if there are any reactions at all. If nothing happens after a week. Then either increase the dose of the suspected culprit or move on to a small dose of the next suspected culprit. Wait a week. And keep this up until you've isolated the offending substance(s).
It is an involved endeavor but because there is a cause that can be known and once known, controlled, it is worth the effort.
The other thing is this has apparently been an ongoing thing with your son so a diet and lifestyle journal is indispensable. If maintained dutifully and accurately it will quickly allow you to correlate outbreaks with certain stimuli and give you the power to guide your son's health.
Best of luck, these kinds of situations where the other guardian is not on board with what would cure the victim of the suffering are very difficult. I'm in nearly the exact same position as my baby sister suffers from bad eczema also and though I know what will cure her, my naive and ignorant, well-intentioned parents would rather pump her full of drugs and heavily processed "health" foods instead of even seriously considering my perspective with 5 years of intense study looking at what has the greatest impact on health.