Did they know you had cats before they enrolled?
I smell a liar here...they probably want to skip out on paying for their notice period.
Even if a child here were to develop allergies to my cats and was pulled immediately, I'd still expect payment. Like any service, I can't help unforeseen circumstances and still need to get paid for holding the spot. All my parents know I have pets and know their kids are exposed despite diligent cleaning to pet fur and dander. They also know of my no refund policy.
You are too nice! I would've just told them that I sympathize with having a sick child, but they elected to utilize the spot despite the presence of cats and therefore they still need to pay for the term notice period. Tough situation...