Politics would be a factor as well over religion itself. If the Khazars compromise and choose Judaism, it doesn't show favoritism to either side. Judaism and Christianity may both be seen as equally incorrect disbelievers by the Caliphate, but Chalcedonian Christianity implies demonstrating some subservience to the Ecumenical Patriarch and Emperor in Constantinople, a factor that the Khazars don't want.
In
Empires of the Silk Road: A History of Central Eurasia from the Bronze Age to the Present, Christopher Beckwith argues:
"The Khazars, the close allies of the Byzantines, adopted Judaism, as their official religion, apparently by 740, three years after an invasion by the Arabs under Marwan ibn Muhammad. Marwan had used treachery against a Khazar envoy to gain peaceful entrance to Khazar territory. He then declared his dishonourable intentions and pressed deep into Khazar territory, only subsequently releasing the envoy. The Arabs devastated the horse herds, seized many Khazars and others as captives, and forced much of the population to flee into the Ural Mountains. Marwan’s terms were that the kaghan and his Khazars should convert to Islam. Having no choice, the kaghan agreed, and the Arabs returned home in triumph. As soon as the Arabs were gone, the kaghan renounced Islam – with, one may assume, great vehemence. The Khazar Dynasty’s conversion to Judaism is best explained by this specific historical background, together with the fact that the mid-eighth century was an age in which the major Eurasian states proclaimed their adherence to distinctive world religions. Adopting Judaism also was politically astute: it meant the Khazars avoided having to accept the overlordship (however theoretical) of the Arab caliph or the Byzantine emperor." (p. 149)