That is, why were the Neo-Hittites of Hatay Province giving themselves a name that didn't resemble "Neshite" but resembled "Hittite" if Hittite was an exonym based on the Hittite control of Hattusa?
"Neshite" comes from the city of Neša, modern Kültepe-Kanesh. Since Kanesh was no longer a significant settlement site, it would've made no sense to continue referring to the city. Additionally, the term was really only used to refer to the Hittite language, which fell out of use by the end of the Bronze Age. The Hittites typically referred to themselves as the "people of the land of Hatti" (LÚ.MEŠ KUR URU.HATTI).
The Neo-Hittites of the Hatay referred to their city-state as Palistin, the etymology of which is unknown. Although the Assyrians sometimes referred to it as Patin, they also used their own term Unqi, an Akkadian form of the West Semitic hamqum, "valley/depression," which became 'Amq in Arabic (Amuq in English).
The Hatay was given its name during the Atatürk reforms in the 1930s, a couple decades after Hittite had been deciphered.