It sounds like the emphasis was on academics rather than on social development in the move from TK to K. It's always a juggling act.
If it's now the end of the year and this problem has been going on for the better part of kindergarten, that's a shame. I don't know where things went wrong, whether the teachers didn't intervene or whether there's some sort of entitlement felt by some private school families regarding their children's behavior. I taught in 2 private schools and saw some of this, although we teachers were on top of the "exclusion" thing like white on rice. It wasn't permitted. So why this continued for a year, I don't know.
But here's the other thing to consider. It's natural to see our children's strengths and abilities, and to emphasize their good qualities. But maybe your daughter wasn't exclusively "mean girled" all the time. Maybe she's lacking social skills and just isn't at the level of the other kids. Granted, they should still be kind and there should be no "You can't play" remarks. But is it at all possible that your daughter reports "you can't play" when in fact the message was that "you can't play like that, rudely or not following the game rules" or something similar? You really have to get info from the staff (teacher, recess monitors, bus monitors, etc.) and not just from your child. If no one noticed anything, that's a clue about your daughter's reporting (and it's normal for kids to inaccurately report). It's also common for some kids to totally misread social cues. If the staff noticed and never told you or worked with your daughter to help her socially, that's a red flag about the school.
The advantage to public school is that the classes are larger, there is usually more than one class per grade (so there is an even larger social group on the playground or at lunch), there may be time on a school bus, and there may be after school activities which offer your daughter a chance to work on social skills and give her choices about whom to play with. In my private schools, where there were anywhere from 8 to 14 kids per grade, you had to play with everyone there, period. Repeating a year would give your daughter a chance to be one of the older ones. As for boredom, public schools usually have various groups going on at any one time, plus paraprofessional aides to help. That lets teachers do a little customization of the curriculum based on kids' needs.
What I think is critical is that you and your daughter not view school as an academic exercise only, mastering facts and figures and vocabulary, but as a place to learn life skills as well. Life skills include gross and fine motor skills, social skills, working in a group, and reading others' body language (who's alone? who's angry? who's sad? who's lashing out but really afraid inside? who's verbal? who's not? who am I annoying with my chatter or my bossiness?).
That applies to kindergarten, but to all grades through 12th. The unhappiest kids in my son's class were those who focused too much on academics and never developed social relationship skills. My son didn't have the highest grades but he got into a better college than his "academician" friends because he had strong social and leadership skills. These came from "reading" others' reactions and needs and being able to communicate and work well with a wide variety of people. He was the kind of kid the colleges wanted because they knew he would function well in a dorm and teams/clubs. So think long-term, don't sweat the small stuff (spelling tests or math facts, for example), and recognize what her needs and weaknesses are. Teach the whole child, not just the brain.
Good luck! Maybe you can find some good readiness activities this summer for her. Do have a solid sit-down with the teachers before the end of the year and, if necessary, ask the school psychologist (even a small private school should have one on call) to sit in on class or recess to observe your daughter and give you some objective data and guidelines.