The ending claims to be describing details "from" (Greek εκ) the Syriac book, indicating that the text itself was unlikely to have been written in Syriac. Moreover, as far as I know this ending of Job does not appear in any other versions than the Septuagint. It was likely written by a Jew, as it adds "And it is written that he will rise again with those whom the Lord raises up" to the last sentence. This is probably taken from a rabbinical teaching, midrash, targum, or other commentary on the book of Job, since it is mentioned separately from the Syriac book. The Jews had no major languages except Hebrew, Syriac, Aramaic, and Greek, and since this text appears in no Targum (again, as far as I know), it is unlikely to have been Aramaic in origin. Thus, having excluded the likelihood of the text being Hebrew, Aramaic, or Syriac in origin, and having confirmed its existence in the Greek text used by mainstream Judaism, it seems fair to conclude that it was written originally in Greek. Such LXX add-ons to books based on non-Masoretic tradition is common (as seen in Esther and the superscriptions of the psalms).
I am far from certain about this answer, but it seems to make more sense than anything else I could find.