Having been in your wife's position when I was breastfeeding I really feel for her. Only my kids both screamed with reactions not eczema.
Email me if you want, I was a pretty restricted eater and had sensitive kids. Definitely the prepackaged foods and the processed foods are out because they have so many ingredients (plus they are notorious for allergy reactions). Are you keeping a diary of what your wife eats? It helps track things down.
Remember cooked oils are difficult to digest, so add the oil after cooking the food. Milk (or whey, casein, etc) is in many processed foods,even a tiny piece can cause horrible reactions. Also it takes (this is from memory it was a while back) 7 days for milk to totally clear from the mom, and then another 7 days from the child (for a total of 14 days). Other allergens were faster to clear for us. Have you been able to go totally dairy free including all the dairy in bread etc?
Since you have a sensitive baby it would be great to get him/her back on breastmilk for as long as you are willing. Please email me private if you wish, I really really feel for your wife, I spent many nights at the end of my rope not knowing what I could eat that wouldn't bother my baby.