It's possible the child has an allergy, but it might actually be something entirely different. Do you know how her parents are taking care of her? Your care might be better. Not saying this to start a fight, just to get your awareness up. Offer to have the baby stay with you again, and see if things improve.
One thing that might help a lot is to give the baby some infant probiotics. They are available at good health-food stores. If you think the parents are not being 100% honest with the doctor, you might want to simply put some in the baby's formula next time she stays with you. It helps the baby's stomach to get the "good bacteria" it needs to be able to digest food. Once it's in there, it should continue to be there.
We had this issue with our first whom is now six.
At home she would be the best baby ever! Only cried when she was wet or hungry. At my mother's on the other hand, (whom had 4 cats that did their business around the house ) she would cry non stop, would be hungry but wouldn't eat, my mother loved watching her of course but too old to put up with the screaming. So we stopped taking her over for a while thought she just liked being home. I ended up having to be hospitalized for a week so my mother watched her. She had issues eating the whole week!!
Longer story short, her Dr said something in animal urin effects their lungs with breathing and causes other discomforts. So the pets were the problem. Told my mother her cats must go . She wasn't able to visit us due to being disabled. So the cats went.
Just let them know their child is more important, and if they are going to gave them they need to care for them better or let them go. Apartment size could have something to do with it? But... we had one cat but cleaned after him often and we lived in a 1 bedroom. Very very I mean very small apartment as it was our first place.
Cat urine can not only adversely affect the baby's lungs, but a very strong odor of urine will affect her mood, even if her lungs are fine. Are these cats using a litter box, and so vigilant changing of the litter would solve the problem, or are the cats just peeing on the carpet and walls and not contained?
I agree with Annie's concerns - they may not be trying as hard to get her to eat as you are, or the problem could be the urine or other concern.