EBF & Eczema - Diet Changes

My LB started getting eczema after his first cold. We got cream from GP which was working but, since his 3rd set of jabs over a week ago, he had a big flare up and the cream doesn’t seem to be helping. I’ve ordered some organic baby cream from Neal’s yard to try. I’m wondering if it could be something I’m eating/drinking that he is reacting to via my breast milk. What’s the first thing I should try and eliminate, dairy? Thanks