At the time of Mongolian conquest of Russia Christianity was quite popular in the Great Steppe. Of course, it was Nestorian Creed but Christianity nonetheless. Sartaq, son of Batu, the 1st ruler of the Golden Horde (of course, the name is anachronistic but let it be), was a Christian but he was ruling for a very short period to be succeeded by his brother, Ulaqchi (presumably, also a Christian), who also ruled only few months. They were succeeded by their uncle, Berke who was an ardent Muslim and there are some theories about his involvement in their deaths.
Now, assume that the 1st thing Sartaq does after becoming a khan, is arranging for execution of his dear beloved uncle.
The 2nd is a switch from Nestorian Creed to the Orthodoxy and a "gentle persuasion" of his nomadic subjects to follow khan's example.
Now, nobody in Russia disputes Khan's authority as a supreme overlord of the Russian lands (Russian Church is eagerly cooperating, being tax free). In OTL the future Islamization of the Horde eventually created "them vs. us" situation but this problem does not exist in this ATL. Unification of Russia happens earlier than in OTL with its center on lower Volga and initial territory stretching from the Lithuanian border to the Western Siberia. Title change from "Khan" to "Tsar" is not a problem: the Russian subjects started calling Khans this way soon after the conquest. The "personal links" were not a problem even in OTL: descent from the Mongolian aristocracy was as honorable as descent from Rurik or Gedemin.
As far as the military strength is involved, "Mongolian Rus" is in a much greater situation than OTL Muscovite state.
How relations with the neighbors are going to develop?