I doubt that the Jewish autonomous republic would be called Khazaria. I am not aware that the Soviets ever promoted that theory about the origins of the modern Jews.
However, such a republic, whatever its name, is not beyond the realm of possibility. Stalin might figure in the mid-1930s that rather than watching Jews all over the Soviet Union for "Trotskyite" tendencies it would be preferable to gather them all where he could keep a better eye on them. Also, Stalin at that time had no ideological interest in slaughtering Jews AS JEWS (real and imaginary Trotskyites were something else); if he felt he could attract people from non-Soviet Eastern Europe who could strengthen his own country he might well have done so. But how many would have come? Many were in denial about the extent of the danger they faced from the Nazis.
Every thing would have become problematic during the Stalin-Hitler Pact period when Stalin fired Litvinov as Foreign Minister so Ribbentrop wouldn't have to meet with a Jew. But would the Nazis at that point have cared if Stalin continued to take in Jews from Poland etc.? At the beginning of the war the Nazis were still speculating on sending the Jews out of Europe rather than killing them (although Hitler already had the idea of killing them, which he expressed elliptically in Mein Kampf).
One thing is sure: if the Jews of the Ukraine and other western parts of the Soviet Union had in large numbers moved to the new national homeland, it would have saved lots of lives.
As to increasing the hostility between the Soviet Union and Israel during the Cold War, I think not. If a million or so Jewish lives were saved, the Israeli government and people would have recognized that fact.
One sinister angle: when Stalin was nearing death in 1953, his paranoia increased and he imagined a Jewish doctor's plot. He began to think of a mass purge of the Jews. What would have happened to the autonomous republic then, if Stalin had survived long enough to carry out his mad scheme.
If such a republic survived the breakup of the Soviet Union I could see it and Israel merging into one Jewish state (this is not an outlandish idea--remember the former East and West Pakistan, also separated by long distances). This would have made the West Bank less important, except for religious extremists, and a way might have been found to settle the dispute with the Palestinians long before now.