My DS has had similar experiences in school. I will go ahead and channel Grinity and suggest the Transforming the Difficult Child workbook. I have used that with some success (and really wish I would have used it at 5). I would also suggest that you consider two issues in dealing with your DD at this point, academic fit and behavior and be working on both of them at the same time. First, do you think that even though she is at the gifted school, she could still be bored? Have you spent a day in the classroom? What about what comes home – do you feel it is at your DD’s intellectual level?

Second, with respect to behavior, the behavior plan that my DS has at school and that has been most effective is to have a daily chart with the specific behaviors you are trying to eliminate – 1) one should be a real problem that you are really trying to fix, the other not as bad a problem and one a gimme behavior. You could meet with the teacher to discuss what are the most problem behaviors 2) For each behavior it is not an all of nothing – it should have 'poor' 'fair' 'good' 'great'
3) State the rule as a negative – for example: No Yelling, No touching, No crying, No moving out of seat while teacher is talking. My DS’s rules are no touching, do not disobey the teacher and no throwing food (the gimme) 4) Leave a blank slot on the daily chart that says: Tonight at the family dinner, we can celebrate that your DD did _______________________.
This part has been great for my DS because no matter what the day is like, we always have something positive to celebrate. It also gets the teacher to focus on something positive every day about your child which can change how she thinks about your DD.
We also use a credit system at home where if my DS gets goods/greats on his chart, he gets 15 minutes of screen time and if he gets a great in not disobeying the teacher he gets an extra 5 points. We give screen time for other good behaviors too. We never deduct screen time but my DS does not watch TV or play videogames or games on my phone without having accumulated screen time.

My DS8 does have a recent ADHD diagnosis and I have found that it is really hard to tease the HG over excitabilities out of the ADHD and vice versa. For instance, my DS has strong traits of 4 of the gifted over excitabilities. Some days I don’t think he has ADHD because his giftedness masks it so well, but one of the hallmarks of ADHD is inconsistency which you mention and my DS has had that too. He can have 3 great days and then a bad day, etc.

Finally, I would just say that your child is 5 and should not be expected to act the same way as kids who are almost 7. My DS may also seem immature by comparison to some of his classmates - he is in a 3/4 cluster after a grade skip - he was still 7 when one of his classmates turned 10.

Good luck. It really is not easy and dealing with people at the schools can be exhausting - hang in there.