If the non dairy formula works and you've concluded that he's got a dairy allergy, I can share some advice for when he gets older.
When our daughter stopped drinking formula and was able to eat foods, we still gave her a bottle of cows milk every once in a while to supplement her diet. For our son we ended up giving him soy milk. A lot of soy milk. I still feel weird about that, I'm not sure if that was the right thing to do, but he turned out fine. We also got a nutribullet blender and made our own "plant milk" for him out of nuts. A handfull of nuts (cashews, walnuts, hemp hearts, etc), about two cups of water, and maybe some fruit (half a bananna, some strawberries, etc), and we'd blend that up and he was happy with it.
I was worried about our son not getting enough nutrition. I ended up getting him some vitamin D drops to put in his plant milk, and if we bought plant milk for him I got the one that had the most calcium and vitamin D. Of our two little ones, our son ended up having the best appetite, he's a big burly boy who loves to eat so i'm not worried about him anymore. Everything turned out fine.
There aren't many effective pharmaceutical treatments for eczema and I didn't want our child to be taking drugs his whole life if he could just fix it with diet. I know it looks like the change in diet is making things worse but I would strongly suggest keeping with it, at least for a few weeks. Good luck!