Given the fact that Chechens wouldn't hurt a journalist who have done his best to help highlight the conflict, and questions asked by the Chechen side about the alleged exchange, we are left with 3 possibilities:
1) He is still alive, but he is being subjected to some treatment aimed at prevention of future objective reports. Alternatively, they just keep him, so that he is not able to contribute to informing the world.
2) Babitsky has been severely mistreated by the KGB, but he is still alive. Due to mistreatment, he cannot be shown, and cannot be allowed to talk to his relatives and administration of Radio "Liberty".
3) He has been killed by the KGB, and a theater of exchange has been arranged to prepare the claim that Chechens have killed him.
Whatever it is, it illustrates once more that Russia is the aggressor in this war.