My granddaughter had similar symptoms. Her pediatrician skin tested her for allergies. She was allergic to milk, soy, peanuts and eggs. My daughter found a formula that did not contain any dairy products. Soy, of course, wasn't an option.
I can't remember the name of the formula she used. I do remember that it is much more expensive than regular or soy formula and can be found at Fred Meyer and probably any large grocery store.
My granddaughter is now 6 and recently saw an allergist. She had skin tests again and of the original 4 she is now only allergic to peanuts and soy.
I am also allergic to peanuts as well as soy. And I'm lactose intolerant. Could be your baby is just lactose intolerant without having an actual allergy. There is no genetic connection between my granddaughter and me. I've never heard mention of soy allergies but here are 2 people who have it.
I have an immediate reaction when I eat something containing soy. It's only hives and an upset stomach but the body's reaction can increase if it's frequently getting the allergic food.
The pediatrician said it was important to do the skin tests, even tho they aren't always accurate, because if the baby is allergic, especially to peanuts, they could have serious consequences if they continued to eat them.
The school nurse said that there are a large number of students with peanut allergies. She said that the number of students with allergies is increasing.
You aren't concerned about peanuts at this time. But she may be allergic to milk or she may just be lactose intolerant. I'd try a non-dairy formula before I tried soy. Or have her tested.
Allergies are not something to mess with.