Since OP has no cost constraints and is interested in such a wide range of majors, I think he should seek out schools where he has flexibility in deciding on a major; those are often private schools. As there is no sense in applying anywhere that doesn’t offer ABET-accredited engineering, that shortens the list of possibilities.
I’d say Case Western (might get in ED), but it is urban. University of Dayton and Marquette come to mind as possible safeties.