What book of the Bible was written in King Tut's time period. (1336-1327 BC)
Answer
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/Tutankhamun’s 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 Tut’s 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 Tut’s time period.
(Correction added 4/2021)Based on the date of the dedication of Solomon's Temple (1 Kings 6:1), 480 years prior would put the date of the exodus about 1448 BC. This would be in the reign of King Tut's great-grandfather Amenhotep II or Thutmoses !V. Thus, during the short reign of Tutankhamun Nebkheprure, the books of Job and the five books of Moses had already been written.