I'm not sure that your choices are binary (home teacher or keep in traditional schooling), based on what you wrote.
She is very young, despite being able to do higher level work. What she may need is less academic type work, and more "fun" but enriching work (which sometimes can be provided in a project-based, Montessori or other type of preschool). She may also need more time to play, dance, swim, do gymnastics or art, etc. Perhaps a nanny would be a good solution if said nanny is good at keeping a highly spirited child appropriately busy with a variety of activities.
It's also possible something else is leading to the tantrums, although some children just have a lot of them at that age, it can also be a sign of other issues, I believe. I'm not an expert there.
