Eczema due to what I eat?

I’ve read that babies can develop eczema as a reaction to a food allergy or sensitivity. I exclusively bf and he’s had eczema since he was a newborn. His doc always said to just put coconut oil on it. Nothing has worked except cerave but even then it barely helps, it just relieves itchiness for a couple of minutes. It’s gotten a lot worse since his last doc visit (4 months) and he has a doc visit tomorrow (6 months).

I’m just curious if I can request an allergy test for him at such a young age. I’d like to know what I have to stop eating so his eczema can go away. I know it bothers him and itches like crazy and I feel so bad.