The First Amendment is pretty clear on this - government can't establish religious institutions, but it can't restrict them either. The whole point of the charter programs is the schools are self organizing and apply for the charter. If they meet the requirements of the charter, denying them access solely on their religious affiliation is almost certainly an overreach. It's the same reason religious non-profits can get access to federal funding for homelessness etc.

This would also not be a novel interpretation - the US tried once but failed to amend the constitution for this very reason with the Blaine Amendment.