Hi J.,
YES. Put her somewhere else and get her started. We had this same problem with both my kids as their birthdays were a few days after the cutoff in September. My daughter (the oldest) was born on Sept. 16, and was the same way as your daughter. But since she was our first, and unfortunately we listened to the advice of our local school system, (Putnam City, who is also VERY unflexible as well!) we didn't start her early. She was bored silly, and then in elementary school there were teachers who would use her to "help" them teach other slower students. NO. My daughter needed stimulation. I went up there and talked to them about it. Not that I didn't want her to help others, I just felt that her needs were not being met. As she got into middle school and high school all of her friends were a grade ahead. She took some accelerated classes (another mistake, but for another time,) and by her senior year, she was ready to drop out completely. It was a huge mess, and one that could have been avoided (in hindsight) if we'd only done what you are thinking of. Putnam City Schools told me at the time that by the time they are in 2nd grade, the age doesn't matter to them, because other people move in from other school systems with different cut off dates, etc. My son also has a September birthday. He was not as mature for his age as my daughter (being a guy) and also the baby, so it was fine for him. But for your daughter, I would certainly go ahead and start her early. My daughter was also small at the time for her age (she was a preemie) but caught up quickly with everyone else. I wouldn't worry about that.
GOOD LUCK!!!
C.