Poor little poppet :(
My daughter’s an adult now, but has had eczema since she was 6 weeks old, so I sympathise completely. (She’s was awkward to diagnose, having allergic reactions everything including the control test patch 😆).
Try cutting out stearates (common in most soaps, emollient creams as it helps emulsify oils into products). We used gallons of Cetreben, only for her Professor (after another long stay in hospital) having a flash of inspiration & shouting “Stearate! It’s Stearates!” to his posse one ward round.
It helped. We used petroleum jelly (and she still does) under wraps/gloves/as a general moisturiser.
Elacom is great as a ‘quick burst’ steroid, along with Fucibet or Fucidin, whichever the antibiotic one is (!) along with Tacrolimus if things are really bad & Icthypaste wraps. A little Aveeno cream rubbed on & washed off instead of soap for handwashing (takes longer, especially if they’ve been digging up earthworms, again). And an old tight filled with cheapo oats in the bath is surprisingly good at soaking off Sharpie, mud, paint etc.
A word of warning. A toddler covered in petroleum jelly is more slippery than Cummings in a rose garden press conference. Buy lots of towel as it gets everywhere & is a useful tool to get a grip on them if they’ve escaped giggling from a bath, freshly jellied & are nude streaking around your house...
Also the usual advice of no fabric softener, non bio detergent only.
I really hope you little one finds some relief. And that you do too.