As I always check when I see these passages, it appears that this lamentation has not been found.
“Laments for Josiah is the term used in reference to 2Chronicles 35:25. The passage reads: “And Jeremiah lamented for Josiah: and all the singing men and the singing women spake of Josiah in their lamentations to this day, and made them an ordinance in Israel: and, behold, they are written in the lamentations.”
This source, as described by the Chronicler, should not be confused with the canonical Book of Lamentations. The same event is retold in 1 Esdras 1:32, although it lacks any reference to writing, or the recording of the lamentation. Nevertheless, the dirges referred to in 2 Chronicles and 1 Esdras, as well as Lamentations may refer to a larger corpus of laments that once existed in the temple or palace archives of ancient Jerusalem.”