Do I keep her home although she insists on going to school to be with her friends, or do I let her continue another year of preschool and risk her being bored? Does it seem appropriate to have her screened a year early and bring this to her school, or do I just wait until next year and see what her public-school screening determines?
It is clear that your daughter is very advanced intellectually, but her behavior has been observed by school personnel to be not as advanced as her cognitive development. My first question, then, relates to the teacher's description of her behavior as being "too hyper." Was the teacher able to tell you what that means specifically? Is the over-activity because she is excited about learning or because she is bored? Second, I'm wondering about the nature and extent of her friendships. Does she have good reciprocal (back-and-forth) play? Does she tend to relate more to the preschoolers or to the kindergartners? The answers to these questions would be strong indicators of which way you might go.
If her behavior indicates that she is excited about learning and if she wants to go to school to be with her preschool friends, then perhaps you should keep her in her current situation so she can continue to learn and improve her social skills and relationships. The teacher might also be willing to provide extra challenges for her to keep boredom at bay. (You would need to make sure that she is sufficiently stimulated and challenged at home as well.)
If, on the other hand, her "hyperness" is actually a reflection of boredom, then continuing in a less challenging environment might be a detriment to her learning and, eventually, to her social development. With regard to private testing, different schools react differently to this. Some find it helpful, others take it as a threat. Maybe you could start by bringing your thoughtful observations to the principal to try to determine a course of action. It is possible that your daughter might be able to go to the kindergarten part of the day so she could get exposure to advanced materials and to preschool part of the day to stay with her friends. If the school is unwilling to make accommodations, however, private screening might provide the kind of data that would convince them. In the meantime, you might want to read Barbara Kerr's book, Smart Girls, which contains some excellent suggestions for parents of preschool girls. Hope this helps!