What book of the Bible was written in King Tut's time period. (1336-1327 BC)
I see you use the late date for the reign of Nebkheprure Tutankhamun. Some scholars date his reign as much as a hundred years earlier. One even dates it several hundred years later. This shows the difficulty of dating the Pharaohs. It is even more difficult to date the writing of some of the books of the Bible with any better accuracy.
Even assuming the dates you give for Tutankhaten/Tutankhamunís reign, and assuming Moses died about 1272 BC (during the reign of Seti I), that would place the exodus from Egypt itself during the reign of King Tutís general, Horemheb (1323-1295). Thus, the five books of Moses could not have been written as early as the reign of Tutankhamun. The only book of the Bible commonly dated earlier than Moses is the book of Job. But that is usually dated much earlier. So, unless one dates Tutankhamun almost a hundred years later than the list you used there were probably no Bible books written in King Tutís time period.