“They seem to be under the impression that the Bible is like contraband in their schools"

As someone who used to teach in public schools, I'm fairly surprised that this impression exists. Most schools I worked in had bible study clubs & the devout among the school population always seems to make a showing for events like Ash Wednesday, etc. No educator I know would be opposed to kids bringing their bibles to school, even those who are very vocal about their non-belief. 

I suspect the impression these kids have is more social than it is administrative. Kids are under a lot of pressure to fit in at school & being openly religious draws attention that most would probably rather not have. It's sad, but high-school kids have weird priorities sometimes. Just looks at how the smart, responsible kids often get treated by their peers simply because they're doing things right.