They will do blood tests when the child gets older, but they are not very accurate on small children (and also children who are breastfed)....
You won't be doing it forever, when your child is old enough to eat 100% for themselves, they can have a non-dairy diet, and special formula or other milk.
You can also ask for a referral from your Paediatrician to a Paediatric allergy specialist.
How do you feel about doing blood tests and then finding out what you already know ? My daughter had lots of allergies, and all the blood test said was 'Oh, yes, she has lots of allergies'...