This is taken from an idea I'm working on for my timeline, which goes roughly as follows:
Emperor Theodore I (no relation to Theodore Lascaris) beats the Mongols and they suffer a painful lesson in humility. The Rus principalities probably still suffer as they're in the way, maybe as badly as OTL. And the Mongols are still the Mongols.
But after Ogedei's death, the Mongols splinter, and the Golden Horde is less of an issue that it was - maybe it never really forms since a Mongol civil war between Batu and the others would be to the east.
So how do the Rus states evolve in this situation? The ERE isn't going to have any direct impact on anything - it has a foothold on the Crimean peninsula but that's it. On the other hand, not only are the Mongols less of a problem than OTL, the Center of Orthodoxy is still around and still a power as time moves on - so no Third Rome is coming in this timeline. But I imagine things will start to divert because of the different circumstances before this.
If the idea for the POD seems implausible, I'm all ears - this is part of a timeline meant to be a success story for the ERE, not a uber-ERE that dominates everything in the East and reconquers the West.
So while I think a severe defeat for the Mongols is entirely possible, crushing them entirely seems absurd. The civil war is just a matter of things working out worse than OTL - judging by most such threats, Hordes are more prone to splintering than uniting, and the possible Great Khans were not on the friendliest of terms.
Sources (in English please) for OTL history involving Russia in the 1200s to say 1500 would be appreciated as well.
Note: The ERE at this point is similar to that of Basil II, though without the Italian presence. The fact eastern interior Anatolia is more "vassal despotate(s)" and less "part of the state" should be noted, but its pretty firmly vassalized.
Using the Byzantine term since...well, I'm writing from a ERE-centric perspective on this timeline, for the most part. That and it seems likely that Turkish vassals drawn into the Byzantine (cultural) sphere will use more Greek and less Arabic.