I read somewhere that SF writers are allowed one “unreal” thing in the world they present, more than that and the ability of the regular reader to buy into it starts to drop. I think it is the same for counterfactual history: you get to make one major change, then follow that to its logical conclusion. My suggestion for your Mongol case is simply: the Mongols sort out their succession by leaving the Great Khan system and becoming The United States of Mongolia with different areas of control and responsibility. Think of it as the US in the 1800s. Ruled from Washington and constantly urging young men to Go West to carve new states in the Union out of the wilderness, if need be by slaughtering the locals.
So many toys, so little time.