I'm not a Barbie fan either. But when you use a gym daycare you kind of take what you get or you don't go. For a regular daycare or nanny you try to find some alignment between your ideals and beliefs, but 90 minutes on an occasional basis? Yeah, pretty much impossible to get everything you want. I don't think it's possible to shelter a child so much that she never even sees or touches a Barbie doll. You still have control over what toys are in your house, right?
Frankly, I wouldn't write a letter. I can't imagine receiving such a letter, if I were the director of a drop-in daycare like at the Y.