If you speak to their admissions officers, you should get a pretty good idea of where you stand. Emory and the Claremonts are private colleges. Admission criteria are usually unpublished and can vary from student to student. Your high school record looks excellent, but competition is cut-throat and your SAT scores are still a bit below the norm. Those scores could be influential.
Some colleges have close working relationships with private college prep schools.
Your high school, however, is more likely to be in the category of parochial, rather than private, and as such will probably not have admission agreements with secular colleges. I'm sure you will do incredibly well. It is a shame his high school limits students in the choice of language courses. I often encourage students to take both a modern and a classical language course in high school. It is also important to consider where you live.