Katharine, both of my sons are allergic to cow's milk proteins. The eldest brought up his feeds constantly. He was mainly breastfed, but I wasn't avoiding dairy and when he had the odd bottle, it was regular formula (and the spitting up was much worse).
My youngest has been diagnosed with a dairy allergy also. He's 11 weeks old. His paediatrician and his allegist have told me not to substitute soya formula for the dairy formula as that might create a soya allergy. He's takes Neocate when he has a bottle. It has no proteins-- built from the amino acids up. Get a prescription if you want to try it, it's expensive. There's another one too. Unfortunately, I completely forget the name of it. It's in a (misplaced) letter somewhere and if I find it, I'll let you know.
There are also some proteins in goats' milk similar to those in cows' milk,. Give them a try but if they don't work, try something else.
My eldest son gained weight normally when he was a baby. He didn't sleep well and brought back feeds constantly, but we thought it was normal baby stuff. It was only after two anaphylactic reactions that he was diagnosed with a milk allergy.
If you want some more info, let me know. My advice is to try the milk-free stuff and see how it goes. If it seems to be helping, give it a few weeks and then reintroduce a bottle with a dairy formula. If you get a horrible reaction, you'll know it's an allergy.
Good luck!