Bartok ended up working for Rasputin because the citizens of Moscow refused to recognize him as a hero
In the prequel, Bartok is shown as a kind, heroic figure, without any of the traces of evil he had in the original (though admittedly, even there he was an Ineffectual Sympathetic Villain). It's hard to imagine, though, how someone as noble and brave as him would end up working for Rasputin, of all people. But the ending to the movie gives us some clues.The climax has a dragon (actually Ludmilla after drinking a magic potion) rampaging through the city, and Baba Yaga (who's regarded as The Dreaded in-story) making an appearance to the citizens. Now, aside from the prisoners, Bartok, and Ivan, not many people actually saw Ludmilla turn into the dragon. The general public probably assumed that the dragon was sent by Baba Yaga and killed Ludmilla. Bartok would have tried to explain what really happened, but no one would have believed him when he said that Baba Yaga was innocent.Bartok, angry that no one in Moscow respects him even after he saved their lives, decided to find a new job as a wizard's assistant. The wizard in question, unfortunately, turned out to be Rasputin.
Bartok never voluntarily chose to follow Rasputin
Alternatively, Bartok might have gotten tricked by Rasputin into doing something that left him magically bound to the evil wizard. That would explain why he never seemed very enthusiastic about obeying Rasputin; he got roped into a gig that he can’t get out of now. When he does abandon his master at the end of
Anastasia, Rasputin could easily have kept him from leaving if he’d wanted to. He was probably too focused on attacking Anastasia by that point to bother with stopping Bartok.
This Bartok is a distant ancestor of the one in the original film, who was named after him
Another possibility is that they aren’t the same character, but are related. This would explain the large time difference between the films. The Bartok in
Anastasia grew up listening to stories about his heroic ancestor. This could overlap with the above possibility, too. Maybe the younger Bartok wanted so badly to become a legendary hero like his ancestor, that he made a Faustian bargain. He sought out someone powerful who he thought could help him achieve his dream, but unfortunately that someone turned out to be an evil wizard who enslaved him instead.
The movie is just an in-universe story
Perhaps it's all just a made-up story that Bartok is telling to his little bat children, he did get together with a cute lady bat at the end of the first movie...
