According to the Ralbag this line is being said in a mocking tone:
ואמר לו על צד הלעג הלא את אשר יורישך כמוש אלהיך אותו תירש כי לא יכול כמוש אלהיהם להצילם מיד סיחון כאומרו אוי לך מואב אבדת עם כמוש וגו'
And he said to him, on the side of the mockery: Will your god Kemosh not inherit it? i.e. Kemosh their god was not able to save them from the hands of Sichon like he said "Woe to you Moav! You lost with Kemosh!"
The Daas Zekeinim (on Bamidbar 21:28 when recounting the experience of the Bnei Yisroel on their travels) similarly brings this out. He notes:
'כי אש יצאה וגו, The entire line is a quotation of how the people speaking in parables flattered the powerful King of Cheshbon. The Torah describes all this in order to show the reader what a tremendous victory Moses had scored when he totally vanquished this empire, annihilating him and his people totally. Yiftach, in the Book of Judges 11,24, refers sarcastically to the claims of the king of Moav in his time, claiming that that the Israelites have to give him back the land that a former king had lost to Sichon.
In a more modern piece, Rabbi Eliezer Berkovitz raises your issue about Yiftach potentially recognising Chemosh as a deity in Man and God: Studies in Biblical Theology (1969):
Even the message of Jephtah to the king of Ammon is consistent with this terminology. The reference in that message to “the Lord, the God of Israel,” who had “dispossessed the Amorites from before His people Israel,” as the English translation has it, may easily be misunderstood as being the counterpart to Chemosh, the god of Ammon. As Chemosh was Ammon’s tribal deity, so was—in Jephtah’s understanding—Y Israel’s. However, if this were so, Jephtah’s plea, also addressed to the king of Ammon, that “Y, the Judge, be judge this day between the children of Israel and the children of Ammon” would be meaningless. The passage suggests that the concept, Y, the Judge, was intelligible to the Ammonite to whom it was addressed. One could call on him to judge impartially between the children of Israel and the children of Ammon. It is significant that, when Jephtah speaks of what God did for Israel, he calls him, Y the Elohim of Israel; but when he calls upon him to judge between Israel and Ammon, he refers to “Y, the Judge.” It was meaningful to speak to the Ammonites about Y, the Judge, because they, too, knew of a Supreme God who ruled over all men. Chemosh for them was the Elohim that was mediating between the Supreme God, the El Elyon, and the Ammonites. For Jephtah, however, Y and the Elohim of Israel were identical. The concept that Y, He is Elohim, does not tolerate any tribal deity beside Y, as it rejects categorically the idea of any mediating divinity between God and man.
So it would seem from Rav Berkovitz that Yiftach phrased it as he did for the Ammonites to more readily understand it. Knowing that they served a deity that ruled over all men, Yiftach was putting it in terms that they clearly understood.