AHC: Make sure that Judaism is a more major religion (much more followers)
Err, the majority of hebrew people still was in Palestina during the Exile. The majors reforms weren't eneacted so you can consider it was Hebrew polytheism/monolatrism, true.Change the balance in power in Judea, like prevent the Jewish leadership to come back from Babylon. This would led to a very different jewish religion.
What is orthodox judaism? At this era, it's largely the judaism that exist in Palestina, that hold Jerusalem. Elephantine Jews are considered heterodoxs because, even if at first they're issued from palestinian judaism, they keeped old usages (as workshipping paedres of Yahve) when Palestinian judaism changed its owns.Don't let the orthodox jew define what is judaism, and let another sect of judaism define it.
Only if you don't pay attention to the large romano-hellen influence on the Christian church. Christianism is as different from Judeo-Nazareism that it is from Islam.Or you could consider that Crhistianism is just another name for Messianic Judaism.
Swedes in their struggle against Denmark read the old testament. Their leadership decides that it is more befitting a warrior culture then the carpenter on a tree.
That might be a bit too ambitious for the Khazars, as that would involve converting the Poles, ect, who have more to fear from Christendom then the Khazars. But it isn't too far-fetched to see most of Modern day Ukraine and Southern Russia being majority Jewish due to the Khazars. No Holocaust may also add about 7 million Jews onto the modern day total, although obviously, the Holocaust wouldn't happen in a scenario if the Khazars are able to leave a lasting mark on the European steppe.That would be interesting. Jewish Scandinavians? If Khazars continued to exist, you might have a Western Europe that was Christian, a Orthodox Balkans, and a Jewish Eastern Europe.
That might be a bit too ambitious for the Khazars, as that would involve converting the Poles, ect, who have more to fear from Christendom then the Khazars. But it isn't too far-fetched to see most of Modern day Ukraine and Southern Russia being majority Jewish due to the Khazars.