There's some great posts here, and this reads to me as either an environmental allergy or a medication side effect. The low dose thyroid meds our Lab is on do make her itchy- but not nonstop, and it took us some time to determine those meds as the source until another vet mentioned it as a possible cause. When she is itchy- which for her (and not all dogs will have the same reaction for the same reason so this is just our case), she seems worse when the weather is more humid. The yeast that causes her itch I alleviate with the cider vinegar soak and tea tree shampoo bath once a week and I bump up her TCM herbals and powdered probiotics- the longest it's ever taken to end was 3 weeks. it took some time to rule things out, but once our arms were around it, if we see the itching or odor start, we take action and end the cycle before it gets too out of hand.
I woul get skin scrapings also- but those weekly baths with proactive ingredients will help alleviate itchiness short term for relief. It is clearly coming from inside her tho, you sound versed in nutrition so the next two ideas I have would be what I stated above.
Please keep us posted as you go.
Human kindness has never weakened the stamina or softened the fiber of a free people. A nation does not have to be cruel to be tough.
Franklin D. Roosevelt