We already have 'choice' via being able to pick any public or charter school. If a religious school wishes to steal public money, then let them become a charter, which is about meeting regulations. 

That is what those universities do to get federal money. 
For example, Notre Dame does NOT require religious classes, though they do make them available.  They do not force religion on their students. It is a NORMAL university.

So, why do you think that private schools, esp. religious, should get public funding, but then be able to do as they see fit, including taking the kids that they want. And yes, many kids are not allowed into religious schools due to handicaps, etc.