You should be able to find a good school that is much more affordable than $200K, but it may not be at a famous “elite” university such as MIT or Princeton, since many of them offer little or no merit money. UChicago is very strong in physics and does offer some merit money, but probably not as much as you’d like. Duke offers bigger merit scholarships but getting one would be very tough. USC, Vanderbilt or Rice might offer somewhat better chances for significant amounts of merit money.
If you’re willing to consider very small schools, check out Harvey Mudd College (which is like a mini-MIT), although it might be tough getting as much merit money as you need from a school that selective.
In addition to the tech/STEM-emphasis schools mentioned above, if you’re willing to consider schools without engineering programs, then also check out:
Grinnell College
Oberlin College
Lawrence University (Appleton WI)
These are all small liberal arts colleges that seem to have strong STEM programs and offer fairly large amounts of merit aid. All are among the top 50 for alumni-earned STEM PhDs per capita (Harvey Mudd is in the top ~5, with Cal Tech, MIT, and a couple other LACs.)
Cooper Union and Olin are two very strong STEM/engineering schools that in the past have awarded scholarships to all entering students (although these have been cut back in recent years and I don’t know how much they’re offering for next year).
I don’t know about the swimming programs at any of these schools.