Challenge: Your challenge, should you choose to accept it, is to totally wank the Khazar state. The state MUST have a Jewish upper class (because, Khazars), but other than that, everything is left open. Can you have the Khazars liberate the Holy Land? if so, what effect would this have of Judiasm?
Well, if you want to
totally wank the Khazar state...
First thing - strangle Islam in the cradle, I don't know Muhammad dying young or something like that.
No, no, wait, the better POD is to have the
Battle of Yarmouk lost by the Muslim Arabs, I mean totally lost - the best Arab generals gloriously died fighting, heaps of Arab corpses scattered all over the place, things like that, you know.
So we have the Muslim Arabs potentially a great threat to the (Byzantine) Roman Empire and the Sassanian Persia, big enough danger to keep their attention and which is more important to keep (Byzantine) Roman and Persian armies closer to Arabia.
And here comes the Khazar Empire - it allies with the Romans against Persia. The Roman Empire after the glorious victory is full of new hopes but realizes that it cannot crush the Sassanians alone. So the Romans get most of Mesopotamia and some territories east of Mesopotamia vassalized just as buffer states.
The Khazar Empire gets the lands to the North of Mesopotamia and actually all the lands around the Caspian Sea, making it "the Khazar lake" and the lands near the Aral Sea, probably some parts of Chorezm. Taking some parts of Persia proper is quite possible, but the main concern here is together with the Romans - not to let the revival of the Persian/Iranian Empire possible.
All the other parts of the Sassanian Persian Empire fell pray to the Eastern (mostly Turkic) entities.
So the Khazars did not get the sweetest chunks, but their borders there in the South are quite secured and they might pay more attention to their Northern borders.
And that means that they truly occupy the territories of the Eastern Slavs probably up to Novgorod, including it directly into the Khazar Empire. That has sense as the trading route "From the Varyangians to the Greeks" is very profitable especially in this ATL when the Khazars have friendly relations with Constantinople and have the route there absolutely secure. The Volga Bulgars are under the Khazar direct control here, unlike OTL when it quickly became nominal. And that makes the Volga trading route even more profitable.
So the Vikings if/when they dare appear there from the Baltics with intentions of conquest will get a bloody nose; but they are most welcome as trading partners of the Khazar Khanate.
So the Khazars will be controlling the richest trading routes including a big piece of The Silk Trading Route from China.
As for the Khazar elites adopting Judaism..., I don't know. It seems that the POD might butterfly this event.
But on the other hand - Why not?
I mean the Khazars don't want to get under too much influence of the (Byzantine) Roman Empere in this ATL as well. So Judaism might get pretty handy - the Empire gets culturally and religiously independent from the Christian Romans.
As for Judaism making the Khazar Empire weaker I am not an expert in this matter.
But it seems to me that idea about Judaism being not proselytizing in the Middle Ages comes from the laws of almost all the Christian and Muslim states punishing with death any adoption of Judaism. Which made Judaism not proselytizing by necessity out of instinct of self-preservation.
Even the existence of these laws in the Christian and Muslim worlds against adoption of Judaism in OTL makes one wonder that these adoptions took place even at the risk of death.
So here in ATL the Judaism without being restricted by the authorities might flourish as a proselytizing religion. Why not?
In OTL Islam in the very beginning was not too proselytizing as well, and the Non-Arab converts were looked upon as "second-rate" Muslims, inferior to the Arab Muslims. But the necessity of the Great Empire changed this attitude with time.
The similar thing might happen to Khazar Judaism, it might get some distinctive features, particular for this branch only, may be.
That would definitely change Judaism as we know it in OTL.
The Khazar Empire might attract a lot of Jewish emigrants from the (Byzantine) Roman Empire, from Africa, Europe and other regions - this inflow of population might be another source of strength of the Khazar Empire.