Ibn Ezra explains it in accord with your reference to the JPS translation:
והיא מלה אחת, כמו עליונים מגזרת גאה
It is one word, like "elyonim" of the group ג-א-ה.
Ibn Ezra seems to be basing himself on the noun form with "-on" suffix that we do find in Tanach - e.g. "pa'amon". The particular word may be unattested elsewhere, but the root is attested, including in a similar lexical context in T'hilim 31:19.
תֵּ֥אָלַ֗מְנָה שִׂפְתֵ֫י שָׁ֥קֶר הַדֹּבְר֖וֹת עַל־צַדִּ֥יק עָתָ֗ק בְּגַאֲוָ֥ה וָבֽוּז׃
Rada"k considers his explanation of "גְאֵי יוֹנִים" to supplement a plain meaning such as this. He says the first word is the nominative(?) plural "g'ei" (similar to the plural noun it was as one word) which is modified by the plural "yonim", as in "harsh mistreaters", or "harsh mishandlers", or "overbearers". This is consistent with his explanation of the related phrase describing a city most understand to be Y'rushalayim in Tz'fanya 3:1 as "ha'ir hayona", which he explains by comparison to Vayikra 25:17, where the verb means "harshly mistreat". This is the kind of activity that seems it could very well magnify the arrogance of the proud people in these p'sukim.
Taking a different approach, Rav Hirsch indeed translates the phrase very similarly to your initial suggestion, with "יונים" meaning doves. His explanation maintains that the theme of the chapter is "the sad present in which the spirit is crushed", but reverses the identification of the גאים as the nation of Israel. This is supported by the fact that the ל preposition after "בוז" usually indicates the object of the ביזה.
"the proud among the doves." As opposed to the imperious eagles which were the emblems of power and majesty of the nations of the world, the "dove," the symbol of weakness and impotence, is entirely at the mercy of all its enemies. Among all the "doves," among all the weak and powerless peoples of the world, Israel alone had the courage and moral energy to stand up with a calm eye to the eagle-like stare of the high and mighty, to remain erect with unbowed self-confidence, and, despite its lowly position, to sense itself as a great force among the national phenomena of world history. The high and mighty dismissed Israel's self-respect as גאי יונים, "the poor man's pride" and they turned away from the Jewish people with contempt. (The Hirsch Psalms, 1978*)
*I'd be happy to include a more stable link to this edition if anyone has one.