Considering that Dooku betrayed her this seems illogical. She wasn't his apprentice for very long. The reason she feels like a Nightsister with the Separatist tag slapped on is because she kinda was. Both in the show and in the game. She also never was a true sith. More an assassin for Dooku that Sidious feared might become an apprentice.
Now Darth Sidious thou. He was Dooku's master and thus, in a way, the true leader of the Separatist. I think he and Dooku would need this kind of synergy.