OK I would respectfully but very forcefully disagree that a sleep deprived mother at the end of her rope should go on an elimination diet.
FUCK THAT.
OP, my son had ambiguous intolerances. Terrible nights, terrible nappies. I went dairy free for a month and it was hell. Do you know how many things have dairy in them? Every fucking thing bar fruit and veg and meat. I was weepy enough to begin with and at the end of the month, after only eating from the Tesco Free From aisle, I went back to my GP, even weepier.
Thank god he very sensibly said "Look, it may not be dairy, it could be anything! Without testing we'll never know, and that referral could take a while. You are not going to eliminate more foods, you need to mind yourself as much as the baby. Go on special formula"
I tearfully agrees, at the end of my rope. He put DS on Aptimel Pepti, warning me that it wasn't the most stripped down formula you could get and he still may not get better on it. He wasn't much better after a month so I went back and we got a prescription for Neocate.
Neocate is the most stripped down formula there is. It tastes absolutely RANK. (Actually most special formulas are fairly disgusting). We went out and bought Vanilla Extract and hey presto! Vanilla Milkshake!
He gobbled it like a gannet. And within two weeks he was a much more contented baby and was sleeping better. And DH was doing half the feeds and I could trot off into town for the morning and get some space without my boobs leaking.
You can get special formula in the chemist without prescription but it's expensive. And it takes a couple of weeks/a month to work.