Yes, you are so right about what you say in the first paragraph. Alexander knows Lyn - and presenting a challenge is part of their usual - mainly intellectual - games. Now, to say the truth, it's only a part of it. Alexander is also very, very proud... And he has his reasons to expect more from Lyn, while not giving up on him.
At the same time, he enjoys rubbing Brad's nose in it, so to speak. But one might just interpret it as a defensive mechanism. Best defense is offense!
I also like what you said last. It was easy to like Brad, after all, when they were young.