Regarding Choate, here are a couple of actions you could take to increase your chances:
One, apply as a 10th grader; you’ve missed a lot of in-person school, so you would probably benefit anyway if you spent three more years in high school.
Two, expand on your experience with Model UN by attempting to improve in various types of debate, as Choate is probably always looking for students who can help them maintain the strength of their team, which I understand is quite good.
Lake Forest is in Chicago, I believe. I don’t know much (really anything) about it. But if you are interested in Midwest schools, you could look into Culver, Cranbrook and Western Reserve Academy.