In Early Christian Literature
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
fruit.
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.
Attempted Links to Ugarit
In light of what appears to be the absence of any other extant extrabiblical attestations to a Ba'al-Pe'or, the rather comprehensive 1999 Dictionary of Deities and Demons in the Bible (DDD)✭ reaches for the next closest thing in its "Baal of Peor" article, viz. speculations on Hebrew etymology in order to connect this deity with the storm-god featuring as the protagonist of the so-called Ba'al Cycle tablets from ancient Canaanite Syria.
According to the DDD (p. 147) the god's surname "is related to Heb[rew] PʿR, 'open wide', which in Isa[iah] 5:14 is said of the 'mouth' of the netherworld" (i.e. She'ol), and thus that the deity as he appears in the Bible "probably represents there the chthonic aspect of the god of fertility... Baal."
The Dictionary goes on (on the same page) to claim that:
A connection may be assumed with the Canaanite deity Baal as known in
Ugaritic mythology. In the cycle of Baal (KTU2 1.1-6) it
is told that in the struggle for dominion Baal is temporarily defeated
by ... Mot, the god of death. Baal has to descend into the netherworld
to reside with the ... dead. In KTU2 1.5 v:4 this is
described as Baal going down into the mouth of Mot (bph yrd). It was
believed that this coincided with the yearly withering of nature in
autumn and winter. In the ritual text KTU2 1.109 we see
that this had its repercussions on the cultic activities. In the
offering list Baal is mentioned among gods who were supposed to be in
the netherworld and who received their offerings through a hole in the
ground (1. 19-23)...
In the Ugaritic text cited in reference to Ba'al being swallowed by Mot (and compared with the enlargement of She'ol in Isaiah 5:14), Death personified is described as gaping so wide that one of his lips sits on Earth at the same time that the other lip knocks against the heavens, while his tongue hits the stars, and that "Ba'al must enter his innards" [or his liver].
Further in the DDD (down the same page):
Num 25 describes the cult of the Baal of Peor as a licentious feast to
which the men of Israel were seduced by Moabite women. In Ps[alm]
106:28 attachment to the Baal of Peor is specified as 'eating
sacrifices of the dead' (Lewis 1989:167). In later Jewish tradition
the cult of the Baal of Peor is related to the Marzeah (Sifre Num
131 and the sixth century CE mosaic map of Palestine at Madeba). In
the OT Heb marzēaḥ is attested in connection with mourning
(Jer[emiah] 16:5-7) and excessive feasting (Amos 6:4-7). So it unites
the different elements of Num 25 and Ps 106:28. This is even more
clear in the ancient Ugaritic texts about the Marzeah, though its
connection with the cult of the dead remains a matter of dispute
(Schmidt 1994:265-266; Pardee 1996).
The Ugaritic text that I can imagine being referenced most strongly here, as a connection between the Canaanite marzēaḥ and the story of Ba'al-Pe'or in Moab, is the one usually referenced as KTU2 / CAT 1.114, in which El, the father of gods (and perhaps also Ba'al's own dad in particular), makes himself sick getting ludicrously drunk at his own feast. (Curiously, a few of the other deities at this banquet are painted in a rather dim light, although the elderly El comes off the worst for wear in the telling of the tale.)
The aforementioned Dictionary article concludes (on p. 148) as follows:
In Num 25:18; 31:16; and Josh[ua] 22:17 the Baal of Peor is indicated
with the name Peor only. This may suggest reluctance to use the name
of a pagan deity. On the other hand, the name Peor with its clear
association to (the mouth of) the netherworld already indicates the
nature of this cult as a way to seek contact with divine powers
residing there.
✭ Edited by Karel van der Toorn, Bob Becking, & Pieter W. van der Horst; Second Extensively Revised Edition; Brill, Leiden • Boston • Köln; William B. Eerdmans Publishing Company, Grand Rapids, Michigan / Cambridge, UK