Something way back in Storm Front got me thinking.
Harry says that you can only have 13 wizards in a circle working together on a spell.
Perhaps the Circle is simply 13 wizards/entities that have got together with a shared purpose.
That narrows down the membership a lot.
Cowl, Kumori, Mavra, Arianna, are all potential candidates.
Interesting that the Grey Council also only has 13...
It's a wonder to me that the Senior Council of the White Council isn't made of 13 Wizards.
Also, Harry speculated that Nemesis might actually be the Black Council (Circle) rather than just an organisation. Jim has said there is a limit to how many Nemesis can control. I'd be prepared to bet it's 13.