Often you find that it is not what you put in handmade soap that makes the soap better, but what you leave out. I would suggest you try a small batch of cold processed soap of a simple recipe with no additives first, and see if the gentleness of that works before adding other ingredients such as oats, honey etc.
My son had really bad eczema from birth and I had him tested to find he was allergic to dairy, eggs, and nuts. Once I stopped feeding him those, it virtually cleared up overnight! Doctors are quick to give you something to put on skin, without looking any deeper into underlying causes. I was living in London, England at the time and the doctor prescribed a hydrocortisone cream for my 3 month old baby's skin. I asked her if she thought it might have been something that I was eating (given that he was being breastfed at that time). She said no, no, of course that wouldn't be affecting him. I went to a naturopath who told of the above allergies (and more), so I stopped eating them and I kid you not, it had an immediate effect.
That said, it's unlikely you find a miracle cure in soap, but you can find/make something that is gentle and won't exacerbate any skin issues your son has. This is usually due to the simplicity of hand made soap and the lack of irritating substances. Unfortunately, melt and pour still has some of those substances.
It should be noted that what can irritate one person's skin, may be fine for the next person - so there may be some trial and error. Apparently lard is fairly well tolerated if you are not opposed to using that. You could go for a fairly basic recipe with only a few ingredients, and if there are still irritations, change the ingredients one by one until you find what's best.
and no more problems. Sometimes doctors don’t know everything, but moms sure do 
