First, I think that presenting the Bible merely as literature reduces it to just another book. England has taught the Bible in public schools for years. Meanwhile, the percentage of people claiming to be Christian has plummeted, especially among those under thirty. Here's the dilemma: teaching the Bible as literature is entirely different than teaching that the Bible as revelation. There is a reason atheists like Richard Dawkins and Christopher Hitchens advocated for the Bible being taught in schools. They weren’t concerned at all about it having a religious influence, because they knew who would control the context. Just to get feel for where this would go in the United States, check out this article from The Telegraph: “WANTED: Atheist to teach religion. Knowledge of the Bible not necessary but experience of asylum seekers an advantage.”
Second, whoever writes the curriculum is going to have a HUGE influence on how the students absorb what they are presented. I think we can all agree it’s not going to be a curriculum from David C. Cooke or Zondervan. It’s going to be from someone like Harcourt-Brace (they make great resources for teaching literature, btw. I used their stuff for years). If David C. Cooke were to make a curriculum about Islam, do you think their bias would creep in? Of course it would, and that's not an insult. It's inevitable. There will be a position taken about the subject matter by whoever makes the curriculum, and that position will influence how teachers and students process the information.
Third, asking someone who is ambivalent or even directly opposed to the messages of the Bible to teach the Bible would be like asking me to teach a class on Scientology. Yeah, it’s never going to get the benefit of the doubt. When I taught an Introduction To The Bible class for a Christian college, I used a Christian curriculum and still supplemented like crazy with resources like The Bible Project. I did this because I had the freedom to seek to make the Bible compelling; I could go into the highways and byways and compel people to come in. This is not what is being proposed in a public school Bible class, and rightly so. A Bible class now is a Quran class later. In my opinion, this is not a can of worms we want to open. 
Fourth, this is going to be difficult for Christian students. Envision a class that claims the book of Exodus chronicles God-ordained genocide. How many middle or high school students can offer John Walton’s or Paul Copan’s arguments against that reading? Old Testament Law? Oh boy. Apocalyptic literature? An understanding of covenants, and why animal sacrifice was important to the ANE? Any concept of cultural context that adds soooooo much explanation to so many confusing things? How the Household Codes of the Romans are echoed and improved in Paul’s writing? Would they be assigned Sarah Rudan’s Paul Among The People or Matthew Rueger’s Sexual Morality In A Christless World before engaging Paul's teachings on sex, slavery, and the role of women in the family and church?
My sense is that this kind of environment will do more to dissuade Christian students from their faith than it will to persuade non-Christian students toward the faith. Until churches all across the United States have a rigorous training program in place for children teaching them to understand and defend their faith, I fear that a class like this will do more harm than good to the enterprise of the gospel.
It is the job of the church and Christian educational institutions to make the gospel message of the Bible compelling. The government’s job is to protect our freedom to do that. If we are deeply unhappy with what our local school is teaching or not teaching to our kids, maybe it’s time to reconsider our educational choices.
 And I guarantee, that can will open. Consider that would happen if a public time of prayer were returned to public schools. The Regent's Prayer will only happen if prayers from any and all other traditions, religious and non-religious, are welcome. I believe there is an assumption that returning prayer to schools will return the Judeo-Christian God to school. That's too shortsighted. It will return every god to school.