One way to maintain a role for psychology, consistent with the answers you have given, is to say that if we have a single psychological successor, then that person is you, but, when you have several psychological successors, none of them are you!
In the original 'body swapping' case Barry has a single psychological successor, the Gary-body person, so Barry is the Gary-body person. In the modified version, Barry has several psychological successors, and so is not any of the Gary-body person, the Larry-body person, or the Carrie-body person. Derek Parfit defends a version of this view. But isn’t it odd to think that whether Barry survives in Gary’s body could depend on what happens to some other, totally unconnected bodies?