I can buy into his transition and I think it's pretty interesting. In the beginning he's so morally uptight and against any idea of cheating that he goes out of the way to unsubtly inform Lynn that someone is cheating off her AND even tells the instructor who doesn't seem to care as much as him. He's bought into the system that told him that if you work hard academically, you can succeed through your own merit despite all the disadvantages and he was confident enough in his own abilities that he felt he could do it. And then the opportunity he's been waiting for happens with the scholarship exam (the principal says it covers from bachelor's to doctorates) and he doesn't even fail but he's denied even the chance. He gets partly guilted into the scheme by Lynn who says, "You're responsible for what happened to me" (i.e. she lost her scholarship opportunity because of him). He notably uses that same line against her later.
I think the common background they share of both being financially troubled super geniuses who were probably the only ones at the school based on merit with struggling single parents probably influenced his decision as well. Like, he knows that she knows how it feels to lose an opportunity like that scholarship. He tells her, "You cheated but I just got unlucky" and he's met with "Even if you don't cheat, life cheats you anyway." After the scrap heap reveal, I'm sure he was thinking about that conversation and how even if he takes the "right' way people with money/power (like Pat) can just fuck shit up for him to suit their whims. There's that moment in the bathroom right before he sends the answer and CEMENTS his status as a cheater, where he almost vomits because it's going against that "honest" person he's been his whole life. and he's like fuck it give me more money because the system i bought into is fucked and the REAL WAY the world works has been revealed ($$$). like "am i going to go through university, exams, the "proper way" to have a CHANCE at a good life? maybe some asshole just ruins it for me near the end just like before. or am i going to take the money right now when the opportunity presents itself. i missed out on one chance before." and when he gets caught, he feels like he has no option but to double down. that's what i got anyway.
I don't have a problem with Bank coming to the conclusion that he doesn't have to follow all of rules if it means bettering his and his mothers life, but his character at the end feels like it's gone too far into the 'money for the sake of money' level of greed. He's Robin Hood if Robin Hood forgot the second half of his motto.
If he tried to recruit her, she turned him down and he left her with a "Well, when you change your mind you know where to find me" kind of line, that feels more like what I would envision Bank becoming. The blackmail feels... too desperate?