They put her up against Angela because she was pretty much guaranteed to win - she was the celebrity guest. It'd be unfair to put a contestant who worked hard up against the celebrity they're supposed to shill. So Cybelle was placed against her as an easy, harmless loss. And by then, Cybelle would have served her purpose and stirred up enough drama.
But Carole and Tuesday not only failed to break up, they're also likely pretty popular for their unique do-it-yourself music and being cute girls. They're marketable and interesting, so you can do more with them. So they bring back Cybelle to stir the pot. Hence why she was allowed backstage after losing.
The "present" Cybelle left for Tuesday was something serious enough to cause massive drama and build suspense for the tournament's darkhorses, but wasn't serious enough to stop them from performing next week. Tuesday had to perform with an injury because the judges knew it would make them more sympathetic. As "compensation," the judges would have moved the girls onto the finals regardless of how they performed. Because there's no way Ertegun would pass them, and Benito was practically salivating over Pyotr.
On top of it all, the mystery of the present would be another story hook. Have the characters turn against each other and fight over it. Make everyone a suspect. Cause drama. In the case of Angela and her assistant, it worked.
- And he uses Angela to get close to her
- Was assassinated by Vanessa
- Divorced Vanessa because she was abusive
- Wants custody of Tuesday
- Is being deliberately kept away from Tuesday by Vanessa