Stony Brook University is well known for both its math and applied math departments. I think it would be at least worth looking into. They offered my son a “scholarship” which essentially made the costs comparable to NJ in-state costs.
I believe the reach schools on your list (Princeton, MIT, Brown) would want to see some sort of math involvement outside the SAT and high school courses, such as participation in the AMC 10/12, AIME or something along those lines.