How terrible for you and your baby! I am now 50 yrs old, and I have had three severe, long term bouts with eczema in my lifetime, and one incident being as a baby. I break out easily with eczema, so I do a lot of things to avoid it.
As an adult, I used capsules of tryptophan and lecithin and that got rid of the eczema within a week. Now, I know you can't find these supplements in a form the baby can take, but, the tryptophan was calming and the lecithin was hydrating, so anything you can find that is all natural to do the same, may be of help. I currently take anti-oxidants to keep the situation under control.
I have sprayed Colloidal Silver on my nephews (I have three of them under age 5 and one is just 5 months old), for abrasions, rashes,and diaper rashes, and they clear up in a day! Colloidal Silver does not sting, and it will kill off the bacteria in the eczema lesions. The two year old takes it orally and in the ear canal when he gets an ear infection, and within hours he is fine.
You can add Colloidal Silver to your dog's drinking water to help kill pathogens such as parasites that can be found in pet dander. You will need to learn how to make colloidal silver (it is very easy) to avoid paying the high cost of it from a commercial source.
Stay away from chlorine at all costs! Get a water filter that will remove chlorine and flouride from your tap water and be sure to use the filtered water for your babies bath.
Check your baby's bedding. Is it treated with fire retardant chemicals? If so, toss it out! They give off toxic fumes that the baby can inhale. Use 100% cotton mattress, then cover that with something that is waterproof and then cover that with layers of washable cotton flannel for softness.
I now use plain old borax (20 Mule Team) for washing clothes and avoid anything that has detergents, chemicals and fragrance. You can add one cup of distilled white vinegar (hydrogen peroxide works good too) to a wash to help kill bacteria and breakdown deposits on your clothes from irritants. Don't burn candles in your home, or spray household fragrances, and get rid of all household cleaners. I put a trigger pump into a 32 oz bottle of hydrogen peroxide and that is what I clean all my surfaces with. It's cheap, works great, kills pathogens, and has no unwanted chemicals or fragrances in it.
I moisturize my skin with organic olive oil, and sometimes use organic palm oil (Tropical Traditions). Both are soothing and healing for my occasional outbreaks.
To learn more about the toxins in your daily life which could be affecting your baby, I would highly suggest reading " The Cure For All Diseases " by Dr. Hulda Clark . It is full of information that is hard to find elsewhere.
I do believe eczema is caused by a parasite that takes up residence in your skin and feeds on different chemicals and toxins in your blood stream. Keeping your body and home free of all the crap that is in modern cleansers, cosmetics and household products, can help you avoid ingesting their toxins through air, water, skin, etc, and therefore not transfer them through breast milk.
When it comes time to stop breast feeding, feed your baby whole foods that you cook and puree. My own son had terrible allergies as an infant and toddler, and when I stopped using store bought baby food and made my own, the issues went away quickly.
I am praying for you. I know how painful eczema can be and my heart just breaks to hear of your baby suffering from this affliction.