Inactions have consequences
"Professor Karkaroff, I appear to have been expelled from Hogwarts School by Dumbledore's refusal to get me out of the competition," said Harry. "As I am without a school, and embarrassed by the lack of support of my innocence in entering my name, may I have the hospitality of your ship until the blasted Triwizard is over? Then I would like to transfer to your school. Your pupils seem like good sorts, which is more than I can say for the Beauxbatons crowd who are as stuck up as Malfoy."
"Harry, my boy, you cannot ..."
"Oh but I can, indeed I must, Professor Dumbeldore," said Harry. "You expelled me, and I find that embarrassing, so I must seek succour elsewhere. I could go and live in Hogsmeade, I suppose, except during the tasks."
"Nonsense, my boy, I didn't expel you," said Dumbledore, crossly.
"Yes you did," said Harry. "One competitor per school. When you agreed with my name coming out, you told me that [a] I am an adult and [b] I am not from Hogwarts school. Had you found a way around it, you would not find yourself in this position. Headmaster Karkaroff, will you take me?"
"If you will protect me from the Dark Lord," said Karkaroff.
"It's a deal," said Harry. "Professor Dumbledore seems to think I'm a weapon to aim at him, and as your school teaches the Dark Arts, I might actually learn enough to understand how to defeat him when I am grown up."
"Harry, you will go dark!" Dumbledore was panicking.
"If it keeps me alive, like I care?" said Harry.
Dumbledore passed out.