Hi there.
I think the best way to test would be to do it gradually, through your BM. That way if it is an allergy/intolerance to pure milk, you will be able to add back hidden and baked dairy, kwim?
So maybe start out by trying something with whey or casein in it. Try a small amout and wait a day or two. Then try again with double what you ate. Then go forward (if no issues like strange stools, rashes, fussiness, etc),. with cheese, yogurt and then full out milk. THEN if no issues try an ounce of formula.
Good luck - I hope there are no reactions!!