1 year old with milk allergy or intolerance? Any other moms dealing with this? Please help?

My daughter developed pnuemonia after I introduced her to cow’s milk a few weeks back. My pediatrician wasn’t sure if it was triggered by the milk or not, or if it was a coincidence that they occured at the same time. I was instructed to give her formula until she was 100% better. I have done this. She is back to normal now. No wheezing or coughing. A few days ago I began to try to introduce the milk again. I started with 1/4 of the bottle being milk and the rest formula. She did fine. Now I’ve bumped it to 1/2 of the bottle milk. After about 1 hour of feeding her she woke up coughing and having the mucusy wheezing again! The doc said if it happened again we could try lactose free and if that doesn’t work maybe soy milk. Has anyone else ever experienced this? What type of “milk” did you end up giving your child? I also wondered if goats milk or rice milk would be another option. Also, her formula is a cow’s milk based, so why doesn’t that cause the same reaction?
Also, we’ve had some allergy testing done via a skin test, and it didn’t show any milk allergy – only eggs.