ADDED: I see you changed the post to reflect your daughter's age. Thanks! Sorry, the original made me think she was three years old-- I get it now. The only thing I'd change below would be my question about her being very young to be in K, but the rest stands. Thanks for the age clarification!
ORIGINAL: If the teacher now says she needs to repeat K, better to have her repeat K now than to wait and find out later she should repeat first or second or third or whatever. The social aspect of that repeat would be much harder on her than repeating K this fall; most of her K classmates in the fall won't know she's repeating and even her church friends who might know she already did K won't care or tease her -- they will probably be thrilled she's there, and she can show them the ropes and feel good about that.
Our godson should have repeated K. He was very, very bright but also disorganized and just not mature enough to move on to first, but despite the teacher's recommendation that he repeat K, his parents said "He'll be so bored academically" and had him go on to first. It took until about the end of third grade for him to really get himself together and for school not to be struggle; had he repeated K and had that extra year to mature, the first few years of school might have been much easier on everyone.
I have to add that I am puzzled by your post because I can't picture any school, public or private, allowing a child of three to do full-on kindergarten curriculum and hours, even a very gifted child. Most here would advise a good pre-K preschool at three and four, with a lot of supplementing by parents (reading, museum trips, extracurriculars). Did I read wrong or was there a mistype that she's actually in K at age three? That means she'd be maybe nine and starting middle school, and there are a host of issues with that kind of age difference between a child and classroom peers that I won't get into here, but.....
Being "precocious" does not equate, necessarily, to being academically advanced, certainly not in all subjects equally at the same time. And being precocious and/or academically advanced does not equate to being mature enough socially and organized enough to move up in grades at school.
I think you may be confusing your child's advanced vocabulary for maturity and her determination to do things her way (butting heads with the teacher, coloring how she wants, etc.) as a sign of something -- giftedness, or again, maturity. Can you distance yourself from her enough to possibly see it as a sign instead of being three and not wanting to follow directions?
You've heard the teacher's recommendation about repeating K, and I find it a red flag in the post that you add how you fear she'll fall behind in first and seem to say the reason is that the first grade teacher is "similar in personality to the K teacher." If I read that right, you're saying you fear the first grade teacher also will not gel with your child. It seems as if you've already written off both the K and first grade teachers at this school, so maybe you really do need to find someplace else for a fresh start. But if she's not ready for first grade at four, she's not ready.
You mention that she would need "a major academic growth spurt over the summer" to start first in the fall. If you know that, you already know your answer: She shouldn't go to first if her success there depends on her somehow gaining a huge amount of academic development over the summer. You simply cannot guarantee that would happen. Why the rush? Not sure how the school or you could justify her starting first grade under those circumstances. Don't let a three-year-old's social worries about friends moving on to first grade sway YOU as the adult.