Heidegger's Bible Handbook: Lamentations: Occasion for Writing
2. Whether Lamentations contains a funeral dirge upon the death of Josiah? The figment of Jarchi concerning Lamentations, that it is that volume which, after the former burning by Jehoiakim.
After the death of Josiah, וַיְקוֹנֵ֣ן יִרְמְיָהוּ֮ עַל־יֹאשִׁיָּהוּ֒, Jeremiah lamented, ἐθρήνησεν,over the death of Josiah, 2 Chronicles 35:25. Which funeral dirge is contained in these very lamentations, according to Josephus in book 6 of his Antiquities, the Chaldean, Jerome, Rabanus, Œcolampadius, Junius, Maldonatus, and others, with yet others denying it, and among thus even Bullinger, Martyr, and Calvin, who calls it a crass error. And certainly in these Lamentations many things are narrated not applicable to Josiah. Neither does the fourth chapter, which the former especially apply to Josiah, square sufficiently with his times. Also, what Rabbi Salomon Jarchi and other Hebrews assert rests upon no reason, namely, that these Lamentations are that very scroll which Jeremiah, after Jehoiakim in a rage burned the former, commanded again to be written by Baruch, Jeremiah 36:32, in which Baruch is said to have written in that book all the words of the book which Jehoiakim king of Judah had burned in the fire, and to have added to them דְּבָרִ֥ים רַבִּ֖ים כָּהֵֽמָּה׃, many words just like those, that is, of a similar argument. But Ibn Ezra and others for good reason refute that. And indeed, in that second scroll were contained all the words of the book which Jehoiakim had burned. But the book burned by Jehoiakim was contains all the words that God had speoken to Jeremiah against Israel and Judah, and against all the nations, from the day in which He had spoken to Jeremiah, from the days of Josiah unto that day, Jeremiah 36:2, which things no one would say are contained in the Lamentations. Nevertheless, that these Lamentations are indicated in the words of Jeremiah 36:32, where many כָּהֵמָּה, like words, or similar in argument, are to said to have been added to the former volume now burned, is perhaps not far from the truth. Whence it is able to be gathered, that the Lamentations were written at the dictation of Jeremiah by Baruch: But whether they were written μετὰ τὸ αἰχμαλωτισθῆναί τὸν Ἰσραὴλ, καὶ Ἱερουσαλὴμ ἐρημωθῆναι, after Israel was reduced to captivity, and Jerusalem was deserted, with Jeremiah then remaining and weeping, and offering this lamentation over Jerusalem, as the Septuagint Translators have it in the Preface of the book, and as the Arab translator repeats it from them, it is uncertain. Certainly nothing appears to prevent the whole Writing from being called Prophetic, since a portion of those chapters speaks of future things, and equally pathetic lamentations of Prophecy and ὑποτυπώσεις/sketches of future calamities are found in the Scripture, even in Jeremiah itself.
 Thus the Septuagint.