If you go court over this, it will be in Small Claim court where each person represent themself. It less formal and Judge able to apply discretion in deciding and not have to be hard letter of law. If you can prove she advertises as pet free place even if not actually in contract, that be good enough. Can also prove it a medical need by simple letter from doctor.

This combined will show you had no option but move your child and if provider made this change without giving enough time for notice, that on her not you. No Judge will expect you pay for choice she made and put your son at risk.

Find any document with her claim to be pet free. Find any correspondence between you and carer that discuss your child allergies. Find anything that support your statement that child cannot be around pet and that this was always pet free day care. Once you have it all, assess if you think that enough to prove your statement of why chose this home and why you need pull child. And then keep it in case she files claim. I bet she won't but if she does, you will have enough showing you had no option.