One possible way to know the language(s) spoken in Kur-Araxian culture is to examine the onomastics in those places were there was continuity.
One such place is the Malatya-Elazig (Kharberd/Harpoot) region that I marked with green circle. In this region the KA settlements evolved into more advanced towns were a type of painted pottery was made until 2000BC. In Middle Bronze Age (2000-1600bc) this tradition continued but new influences from South appear with Khabur ware while close to 1600BC influences from West, from plain Anatolia become stronger.
After 1600BC the region was incorporated into Hittite empire and became known as Ishuwa.
The main ethnicity of Ishuwa is unknown. They are no native texts. Three kings are known from last periods, all have Hurrian names and based on that some proposed that it was predominantly a Hurrian place.
Nevertheless king names are not always a good indication of real ethnic situation. Mitanni had Aryan kings but was a Hurrian kingdom. Hittite empire had kings with Hurrian names yet it was still Hittite. Babylon was ruled by Amorites, Kassites and Chaldeans yet it's main language was East Semitic Akkadian derived. Such examples are numerous.
For this reason Aram Kosyan (based on Laroche's etymology) examined the names of local Ishuwan chieftains, elders and rebels. Surprisingly the most frequent group consists of Hittite-Luwian names. There is only one Hurrian name Kumarpi, few Kashkean and Indo-Iranian names. There is also one name Mita that looks like Phrygian name but a presence of any real Phrygian in that region is unlikely. A similar to Mita name is attested in Armenic Etiuni: the Muida. Given that this name appears in Iron Age as Mita of Mushki so the most parsimonious explanation is that it's related to Mushki, and maybe Etiuni. Although a Luwian etymology also exists but in my opinion it's problematic.
Anyway returning to main subject, it's somewhat unexpected to have such result. There can be two explanation.
1. Either those Hittite names were there since Kur-Araxian period.
2. Either they appear with Hittite conquest.
We don't have ancient DNA from Middle and Late Bronze Ages (2000-1200BC) in Ishuwa region but we have a sample dated at 2400BC from EBA. It's ART001. It's more southern shifted than other three samples from EBA period. This extra Levantine shift can indicate that Hurrian speakers infiltrated there later from South which opens a gate that Kur-Arax in that region was really speaking Anatolian IE languages.
This unexpected result is further reinforced if we take into account that the kingdom Armi dates (before 2250BC) overlap with Kur-Araxian culture in the Cilicia where it was located according a recent paper. Kroonen suggested a strong Hittite presence in Armi based on Eblaite inscriptions.
Another potentially IE term Aratta also coincide with Kur-Araxian period. It was probably located somewhere in North West Iran, where in Iron Age a river Aratta is attested.
Offcourse we must wait ancient DNA to further confirm or dismiss this theory. Also we must keep in mind that Kur-Araxian horizon was quite large and harboured other languages. There is little doubt that Nakh-Daghestani in North East Caucasus descend from KA. Also Hurrian languages were probably spoken in south of Van, Hakkari region KA settlements. In any case it seems that the main expansion of Hurrian occured in MBA with Khabur ware.