Wow. Definitely a dilemma. I teach at a high school and we are a big hang out place before school - BUT, the kids aren't checking out books and bringing them back unread as a means to visit the library before school.
Have you spoken with your principal or asst. principal about the problem? That would be my first step if I were in your position. Perhaps he/she can offer some suggestions or address the problem with the faculty so that you don't come across as being "the bad guy."
One other step that might work: do you have student library helpers? Perhaps some students from the 4th and 5th grades who are responsible and could be taught how to correctly shelve books would be able to help you.
Another possible approach, but it could cause just as much work: Since students are wanting to check out books daily, could you give the teachers who seem to be sending them the most baskets of library books to keep in their classrooms? The books could be changed out every two or three weeks - solving the problem or students coming into the library and not reading the books. Maybe this way, some titles that might not be noticed by real readers might get read as an added bonus.
Wow... sounds like the teachers at your school aren't very interested in collaborating with you. Would it be possible to assign each classroom a day of the week that their students could come for open checkout? Like Kindergarten on Mondays, 4th & 5th on Tuesdays, ...etc? Another idea could be asking for parent volunteers who could come in on a regular basis to help with the re-shelving.
Either way, it definitely sounds like you need to visit with your building administrator to get them on board though; It sounds like you are being taken advantage of!