Consider that when you look around at other happy, productive adults, you can't tell if they attended kindergarten or not, or if they started young or were held back. You can trust your instincts that your daughter will not be scarred for life by your careful decisions regarding her early childhood learning experiences.
My son, too, had an unpleasant start to his kindergarten year at our neighborhood public school. After Christmas, we chose to pull him out and had him finish his kindergarten year at a small Montessori school. It was the best thing we could have done, for a variety of reasons. The teachers were gentle and responsive (the previous kindergarten teacher had been a big part of our disappointment) and our son just thrived in the mixed-age classroom.
Montessori was not a long-term commitment for us; our son went on to attend first grade at the local public school the next year. But the time he spent in the mixed-age-group classroom was so valuable. The kids ranged in age from 3 to 6. He could look up to the children who were older and more advanced, and was encouraged to help the younger children. Academically, between me having fun working with him at home and all the wonderful things he learned at his Montessori school, he was well-prepared for first grade and was and is still at the top of his class. If you have a nearby Montessori school, I'd encourage you to look into it.
You could also find some homeschooling families and get together with them for group activities for socializing experiences, or just enroll your daughter in a couple of fun classes (dance, gymnastics, library story time, etc.) and work with her a lot at home so you feel confident she has learned what she needs to, both socially and academically, to start first grade.
Personally, my birthday fell the day after my elementary school's age cut-off day, so I started preschool as the oldest in my class. Then, in kindergarten, I could read fluently and easily and was bored learning letter sounds, so I was put into first grade, where I was the youngest in my class. I stayed there until I reached fourth grade and *struggled* with the math concepts taught. Looking back, I believe I was just not ready to understand some of the abstract concepts, and the teacher was not one I particularly connected with. I repeated fourth grade and was back to being the oldest in my class. This is the age group I stayed with until I graduated from high school, so I was always one of the oldest in my class. No big deal. I turned 18 at the start of my senior year and it was really no big deal.
Best wishes!