I wouldn't feel that you shouldn't apply to certain schools because they aren't non-denominational. Most schools tend to have episcopal roots, however there are students of many religions at these schools. Some of these schools go to "chapel" however it is more of a morning meeting where they discuss friendship or honesty. They also may do some announcements or give awards. I know this is how st. paula does chapel.
I also want to apply again to a school I was rejected at, however im not sure. I read somewhere that 75% of Andover applicants are academically capable of the work, however it has around a 19% acceptance rate. Let's then assume that 5% of applicants are waitlisted. That means around 51% of qualified applicants are rejected at Andover. I assume that these numbers would be quite similar at other top schools, especially Exeter. You may have been in the 51% rejected but academically qualified, and for this reason I would apply again. In the interview, talk about how it makes you determined and you stick to your goals.