Thank you so much for your email. The piyyut is from medieval Italy, probably 12th or 13th century. (No earlier than 11th, because it shows clear influences of the late-10th-century "Odkha Ki Anafta", and no later than 14th, because it appears in manuscripts from then.)
It occurs to me that the piyyut is probably later than "Ahallel El Beminnim", another Hanukkah yotzer, because "Ahallel" also begins with Alexander. The difference is that "Ahallel" is based directly on the historical narrative of Josippon, from Alexander's conquest through the victory of Judah Maccabee. The Alexander section of "Ahallel" tells the story of Alexander and the High Priest, like our yotzer, but then it goes on to tell the stories of Heliodorus, the translation of the Septuagint, and finally (the bulk of the poem) the various events associated with Antiochus IV and the Maccabees.
If our paytan was inspired by "Ahallel", it means he was so enamored of the Alexander section that he decided to devote almost the entirety of his piyyut to Alexander, and only mention the Maccabean victory very briefly, at the very end. This means that he "recasts" (to use your term) the story of Hanukkah, or, at the very least, shifts the weight of the holiday to Alexander.
This is interesting, because we know that Alexander was a popular character in medieval European literature in general. Usually, when we think of Alexander in medieval literature, we think of the Alexander Romance, which was available in many languages, including Hebrew (where it found its way into later versions of Josippon). However, the stories told in our yotzer are not those of the Alexander Romance, but of the Talmud and Midrash. Perhaps further research (not mine, but I'll make an appeal for such research in my publication of this yotzer!) will uncover further evidence that Jews associated Hanukkah with telling stories about Alexander. (Or, on the other hand, perhaps not -- it could have been the creative innovation of our paytan, who wanted to make Hanukkah about Alexander, but did not succeed in spreading this idea.)