The only ancient sources I have been able to find are from the works of Roman and Greek Christians from the early stages of the Christianisation of the Roman Empire. These typically are acquainted with the god's name via the Ancient Greek translation of the Tanakh (commonly known as the Septuagint), in which he is called something like Beelphegor.
Moreover their main concern with the topic of his name is the performance of some interpretatio romana and interpretatio graeca, i.e. their attempts at identifying a foreign deity with a Roman or Greek one with whom, in their own homelands, they would be more familiar.
In Ch. 2 of his Commentary on Isaiah (writing in the late 300s & early 400s AD), Eusebius Sophronius Hieronymus, better known in English as St Jerome, says that "the idol Beelphegor" is to be interpreted as Priapus. According to Theoi.com, the Greco-Roman divinity Priapus
was the god of vegetable gardens. He was also a protector of beehives,
flocks and vineyards.
[He] was depicted as a dwarfish man with a
huge member, symbolising garden fertility, a peaked Phrygian cap,
indicating his origin as a Mysian god, and a basket weighed down with
In Against Jovinianus 1.22, Hieronymus highlights further the logic behind this identification by appealing to an apparent similarity between the Greek/Latin word Priapus and the "Hebrew" Phogor, i.e. Pe'or.
According to p. 74 of A Dictionary of Angels, Including the Fallen Angels, by Gustav Davidson (The Free Press, A Division of Simon & Schuster Inc., 1967, New York, NY), Rufinus, a contemporary of Hieronymus, also identified Beelphegor (or Belphegor) with Priapus.
According to p. 97 of the Rev. John Thein's Ecclesiastical Dictionary, Containing in Concise Form, Information Upon Ecclesiastical, Biblical, Archæological, and Historical Subjects (Benziger Brothers, 1900, New York, Cincinnati, Chicago), Origen of Alexandria, who lived a couple of centuries earlier, made the same identification.
In his Commentary on Psalm 106, Theodoret of Cyrrhus, yet another contemporary of Hieronymus and Rufinus, says that "Baal Peor is an idol" while "Peor is the name of the place of the idol", and also that the idol Baal is "called Kronos [Cronus] in the Greek language."
The 10th century AD encyclopaedia called the Souda [Suda] seems to be referencing Theodoret when it uses almost these exact words of his for its definition of this god's name under the encyclopedia's "Beelphegor" and "Phegor" entries.
Thein's Ecclesiastical Dictionary says that Apollinaris and St John Chrysostom also identified Baal-Peor with Saturnus, in which case the Roman name for the Titan Kronos is being here employed.