A verbal agreement can be binding under the law. In order for a contract to be binding, there needs to be 'consideration' - that is, something given in exchange. A mere promise to do somerthing, for no reward, is generally not enforceable. Anyway, this sounds less like a 'contract' (which is typically a sale or exchange of goods or services), and more like a settlement agreement. Those too can be binding, but again, it would have to be in settlement of something, as part of a mutual agreement. The presence of attorneys is irrelevant.
There's a more fundamental question that occurs to me: If your daughter doesn't want to change schools, are you sure you want to try to force her to? There are sometimes important reasons for parents to override their children's wishes in matters like that, of course, but surely you want to grant your daughter's desire some weight.
I don't practice in Wisconsin, so this isn't meant to be legal advice, just my reflections. You should consult with an attorney who practices there if you want to try to enforce anything.
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It isn' t binding but as joint custodian you have the right to have an equal say in schools. Unfortunately, the courts tend to favor the status quo. Since you have or had an attorney, you should discuss your options with him/her.