I actually know what this is or at least I have an idea.
Go to your doctor and ask to about a nickel allergy. The placement of the rash is in the right spot. People usually get them on their feet, earlobes, or the side of the finger. I had it on my right index finger and it was awful.
It's mainly treated with diet, e. g. you stop eating canned food and any number of other foods that have nickel in them.
It might not be this specific allergy, but it does sound like an allergy of some sort. Best to have it checked by a physician.
PS My own doctor had to look it up in his medical textbooks so this might be why the specialists you've seen have missed it. I'm fine now; the diet works and I only follow it strictly when I feel the symptoms returning. I also stopped wearing earrings altogether, as that seemed to be aggravating things (of course you don't wear earrings on your hands, but the hand is one of the places where this allergy seems to manifest itself, regardless of where the exposure to the allergen really was).