Hanukah has Megillas Antiochus.
It is mentioned by R. Shimon Kayara, a Geon, in Halakhot Gedolot, and Rasag attributes it to the Maccabees themselves. The geonim held in high regard and believed it to be authored by Tannaim of Beis Hillel and Beis Shammai.
Italian synagogues would read this on Hanukah, and in the Baladi nusach of Temanim, they still read it as part of the davening. R’ Yahya ben Yosef Zalih, in 1715, wrote that this megillah was read after the Haftorah on Shabbos Chanukah, and he considered its reading an important part of the mitzvah of Pirsumei Nissah.
The Machzor Kaffa, published in 1735 in the Crimea, says that this should be read during Mincha on Shabbos Chanukah after Kaddish Titkabal.
I see no reason why this text couldn’t be recited on Shabbos Chanukah, either during Mincha or Shacharis (similar to Koheles). Obviously, there are no brachos, but I don’t see how this could be a problem.