Another possibility: she asks for a bilingual school so she can do academics in both languages. She wants to play at school because the social scene there is what she hasn't mastered yet, and she mulls over the 'fights' for weeks because she can't figure out the other kid's motivation. In contrast, at home she knows how the social scene works and is ready to pay attention to solitary pursuits.
If this rings true to you, then she may need coaching to understand her agemates, and much much more academic challenge. School may be the right environment for her to learn about other kids, and home may be where she studies, at least for now.
