He didn't convince her to stay in the academy. He told her what was going to happen and allowed her to make an informed decision. He loves his daughter, but he described her as "reckless". I'm assuming that he had met Sophie several times and respected her. He knew what Kate was going to do and that she was going to talk Sophie into doing the same thing. He presumably knew what her career hopes and dreams were and how much they meant to Sophie, as he would be someone she might ask advice from.
He told her that she had to choose between her relationship and her career. He just pointed out that she couldn't have both and she would have to make the difficult decision of which one she wanted to chase. One could read that conversation as him making her aware that he would help Kate (and, by extension, Sophie) if he could, but that none of his influence would change what was about to happen. Maybe he thought he could predict it, but he didn't know what was in her heart. He didn't know how she would weigh things. Only she could make that decision for herself.
This is for the benefit of Kate. If Sophie follows Kate out the door, but would have stayed if she understood that she had no chance of winning, then she would have grown to resent Kate and their relationship would never have survived that resentment. If Sophie, after a careful consideration of her options, still wanted to go with Kate, then I think he would have supported them. He wasn't telling Sophie what to do. He was asking her if this was really what she wanted.
He did something that was in the best interests of both Kate and Sophie.