I lived with burning feet for years so understand your misery. You might consider you have multiple issues causing burning feet. Often with autoimmune issues it's not simply one issue but multiple issues causes symptoms. I had burning feet starting in middle age: my foot pads the most afflicted and the redness, burning and pain much worse in the evening; trying to sleep was difficult. Could you also have allergies to contact materials? Were any contact allergies ruled out with the 5 Day Extended Patch Test? This is not the simple 30 minute test for environmental allergens, Google information about it. This 5 Day Test test revealed reactions to over a dozen daily contacts. To stop the burning feet I needed to create a barrier or walk on allergen free surfaces. I can walk barefoot on dirt or sand. But even barefoot inside on surfaces safe for me the soles of other peoples shoes contain an element from rubber which contaminates the floors. Or the floor is cleaned with cleaners I'm allergic to. The only shoes safe for me are 100% plastic, vinyl or leather contacts. Otherwise, in a tennis shoe for example because of elastic allergies I wear a 100% cotton sock; Buster Brown and Cottonique online, both sell elastic free, 100% cotton socks. I then put on a plastic bag and cover that with sock containing elastic to support this layering. I can then wear tennis shoes or shoes of other blended materials. Yes my feet get hot, but in my case that doesn't trigger my burning fee. In your case that would be an issue. A cotton sock alone will not provide a barrier from surfaces you are allergic to. There are a few 100% vinyl flip flops and shower shoes , but EVA, Ethylene Vinyl Acetate, is replacing the 100% vinyl and I have found one brand that seems safe so far, yet haven't used long enough to be sure because I but have bought other EVA sandals and my foot pads begin to burn again. The expensive 100% leather shoes are supposedly safe for me, as I didn't test positive to formaldehyde for example which is often used to tan the leather, yet some brands make my feet feel hot, not to the burning point, but I only wear these shoes for a few hours at a time when I need a specific style for social situations.
I hope you can find some relief.