Like a previous poster said, Abrahamic religions are difficult (but not impossible) to dislodge.
Maybe you could have a situation where a primarily Kalmyk horde establish dominance over Muscovy, and institute a policy where only Buddhists can hold higher public offices. Mix this with a Janissary-type policy where they take Russian boys and educate them in more martial interpretations of Buddhism, whilst breaking the social power of the Orthodox church through a mix of persecution and structural suppression. Replace them with the Buddhist temples as financial power centres and the Buddhist minority will call the shots.
Plus if any Orthodox-Russian bandits/ dissatisfied locals attack the temples, they can be repelled by monks who have trained in both Eastern styles of martial arts and traditional *Russian and Caucasian wrestling and fist-fighting styles.