My brother wound up in a similar situation. Davidson County had "lost" their record of him paying some fine or another and demanded that he pay them a second time.
They have three metall drop-boxes outside the building... I put the book in one of them after the library was closed. They cut you off after your late fees exceed $25.
That happened to quite a few people that used the drop off slot at this one small video store in town. It was a scam and word got around very fast.
Go into the Library, check to see if your book is back on the shelves. find it, take it to the desk and tell them, with a smile that they had your book all the time, you understand how these things happen and REQUEST, don't demand, that all fines be dropped. It would not hurt to have a stranger see you find the book and accompany you to the desk.
Keep in mind there may have been a double check out also. All too often when the staff comes in to work they gather the returned books, place them on a rack to be gone through at a later hour. People coming in early morning may examine the returns and find one they like, walk in and check it out before the staff has time to credit you with the return.
You can request a history of dates and times the book was checked out and returned, and find the problem with an overlap in its history.
Well worth your time and effort when it comes to that much money.
With problems such as yours if you work with them they will be overjoyed to help you and themselves find the book if possible.
Please let us know how this turns out. Regards Vesta