Let me just put this out there: People love their pets unconditionally and think that anyone who steps across their threshold should adore them as much as they do. So it's not a stretch that your MIL would meet a request to "put away" her cat as warmly as you would if company came to your house and asked you to "put away" your daughter. Ugliness or at least some passive-aggressive behavior will be exhibited, but it has nothing to do with how she feels about your kid. It's all about the cat.
Hand this over to the pros. Bring her to your kid's next doctor's visit so she can hear firsthand that her cat is making her grandchild sick and that you're not being high-maintenance. Then ask the doc for tips on reducing the allergens. They may include things I do before my dad, who's allergic to my dog, visits: I groom Teddy, we use a special vacuum that reduces dust and allergens in the air, and we don't allow Teddy in the guest room ever, so Dad has a safe haven.
Try to get your MIL to go with you and your daughter to the park or another place where the cat isn't an issue. After all, the most important thing is that your daughter get QT with her grandma -- and that can happen anywhere.