You mentioned that you recently moved. Have you had the air ducts/vents/furnace professionally cleaned? How about the carpets? Some pro carpet cleaning companies like Stanley Steemer say they are able to check for allergens, I believe. What about the padding under the carpeting? Has it been checked? Previous owners may have not been scrupulously clean or might have had pets. Carpet padding can be horrible.
As Wild Woman said, water could be a concern. My daughter has recently been diagnosed with Aquagenic Pruritis and Aquadynia, although not Aquagenic Urticaria. She suffers from unimaginable itching and pain after a shower, or when she sweats. One possible thing that can help with any of those conditions is to coat the skin with a neutral oil (like pure coconut oil) prior to the bath or shower.
Also, have your water quality checked. They might be adding a lot of fluoride or other substances in your town. You might have very hard (mineral rich) water and he could be sensitive.
And did you have the mold checked by a specialist, or did you do a visual check yourself? A pro might find something you can't.
And you probably know this, but you mentioned eliminating fragrances. I'm assuming you know that unscented and fragrance-free are different, right? Unscented is not fragrance-free; they add a fragrance to mask any scents.
By the way, they recently discovered that the Honest Company soaps were adding a substance that they say they don't add. It was a clever word trick, basically. I'd ditch the Honest Company products, and instead only use Cetaphil or Vanicream or something that a pharmacist can recommend.
I'd also suggest trying something other than Biokleen. Try baking soda only, or try a different allergen-free detergent.
Good luck. I'm the mom of a medical-mystery child and I know it's frustrating. Don't stop looking!