I think it sounds like a great idea... One question though: have you tried going to the school and asking for financial aid? Sometimes they are more accommodating to children like yours because they want the scores from the child included in their averages. Maybe a sort of academic/need based scholarship combo. On average private schools tend have about 30% of students on some sort of financial aid and some kids can get full rides (this can vary widely by school though but I keep seeing about 30% over and over).
If you have decided homeschool would be a better option anyway:
Remember she will move faster through the material AND that you have two summers to use if you and your daughter choose to use them. Kids often skip and its a history class/social studies that just get skipped all together for the year. No problems. Honestly she will see it again, particularly in a public school (they continue to revisit topics that are important). Also, most of the material that is important from those classes could be taught in a few focussed months... what you do miss is the interesting conversations, projects, "public" speaking, research papers ect. that do build upon each other to prepare you for future classes. She could be fine skipping these if she is already strong in those areas, particularly writing.
