You made two mistakes in your post.
1- The fact is that the public schools are essentially just that public - which means government controlled. So it is not a question of inviting. They are the ones who own it.
2- The second point you miss is that Tremoglie says that they are invading prviate rights against religious expression in public.
Read it again.
What in your education of America's history and her Constitution allowed you to conclude that the federal government owns the public school system? We the people established a government with only the specific and clearly enumerated powers outlined in the Constitution. Owning, controlling, or funding, the public school system was not one of those powers.
Congress which created the Department of Education in October of 1979. The question I posed was where in the Constitution was Congress granted the power to create a Department of Education?