She is unhappy at this school. You said it yourself. I would seriously think about alternatives. If she is going full time, school is a major part of her life. If she wants to play at school, that is pretty appropriate for a three-year-old regardless of what she can do at home. Perhaps a smaller school that is more play-based and nurturing of interpersonal relationships would be appropriate right now.
Her poor behavior could simply be a result of an inappropriate school fit. If she is still throwing huge tantrums at almost four there is something going on. It may be that the bilingualism stresses her out. It is really interesting that she begs to go to an English speaking school. It sounds like she feels like she does not fit in at this school which is heartbreaking.
I would not worry about an academic fit. Preschool is supposed to be fun. I suppose an age-based school where you are would be a lot of playing and pre-literacy stuff--nursery rhymes, story telling, songs etc. That might be just what she needs. She might really benefit from that stuff in her second language. And, the age-based schools might be plenty challenging as she navigates learning to read and write well in two languages.
We are currently making choices for school next year for our sensitive and shy, reserved child, so I know it is tough to really see what environment is the best fit. FWIW our DD just wants to play at school. When we talk about kindergarten next year she does not want to go because she thinks it will be too hard. This from the 4-year-old who reads at a second grade level and is probably around that level in math too. She loves learning about history and science at home but loves playing with friends at school more. It has yet to be a problem just letting her do that. Good luck.
