They say they aren't capable of wiping out the enemies, but shouldn't Anko be capable of that by herself considering she's a max level secret endgame boss?
I'm also sensing the first prince will be the one to drive the final knife into the chancellor's side as he realizes what's going on.
I'd assume Anko's turn will come in the battle against the Halidom forces.
It might be better to allow the second prince to seemingly defeat the chancellor's side in a more traditional manner, rather than allow a monster connected to Second to do it. The second prince being there, along with mainly kingdom soldiers, would give a whole lot of legitimacy to the second prince once he becomes the king. If someone else, like Second & Co, visibly handles it, nobles and people would wonder if the second prince really is his own man, not just Second's puppet. You might be onto something with the first prince finishing off the chancellor, though. Maybe he won't even try to become the king after that, feeling himself already disqualified after having teamed up with a foreign infiltrator.