August knew how meticulous Ghost Flash was. He would normally always think about something before putting it into action. That was what August knew about him and hence left him in charge of seeking new members. He did not rely on Ghost Flash's judgment alone. He also had his means.
Throughout his conversation with Claire, he had Rhea analyze her and he was constantly fed with data on her heart rate, breathing, and her minute reactions. He even deliberately touched her every now and then to increase the effectiveness of Rhea's scans. Of course, he indeed tried to comfort her but he also subtly verified that she was not a spy. He never let down his guard.
At this moment, he knew more about Claire's present situation than she even knew about herself. That is why he decided to strike the iron while it was hot. Having verified for the most part that she was indeed genuine and knew all about her fears, he was confident that he could win her to his side with his offer.