My SIL was on a special diet with her son because of similar situation - skin rashes /eczema etc. not too sure about screaming.I don't remember but I know he did cry a bit.I want to say you are doing a fantastic job and send you a big hug. It is distressing to see our babies in pain and for us not to be able to help them.
Let alone sleep deprivation.
I will have to ask her Dear nephew is now 4 so it isn't too long along.She continued to breatsfeed but I know she stopped eating potatoes an dlots of other things as he showed an allergy/intolerance to it.
I hope you have continued success and your baby settles.