You propose taking out loans for “the rest” as if it were just a small gap. 40K/year would cover only half the cost of attending an Ivy. Even if the shortfall did not interfere with getting a student visa, it would leave you and your family with a six-figure debt, for an undergrad degree. This would have a very significant impact on your life.
There are many excellent colleges and universities where you could get significant merit aid, and/or where the “sticker price” is lower.
Harvard, Yale, Princeton, Amherst, Dartmouth, Bowdoin, and MIT are all need-blind (and full-need-met) for internationals. If you have demonstrable financial need, refraining from applying for financial aid at any of these would be exceptionally foolish. As an international applicant, you’d essentially be incurring the disadvantage of competing for that rare international aid without reaping the benefit if you do get in.
Obviously you have an extremely strong record. I hope you’ll make all that hard work pay off by applying to schools that will reward your achievements either with merit scholarships, or by accepting you with need-based aid even though they’re need-aware. (But of course, the latter can happen only if you apply for aid.)