Input from a teacher:
If she's not extremely (and I mean extremely) bright, has grown up in a small town and attended a not very academically challenging public high school for two years, she'll probably struggle at a college prep boarding school. Boarding schools aren't filled with geniuses (there just aren't that many geniuses around), but they do cater to academically competent kids who are from competitive school backgrounds. It doesn't sound like she's been in a competitive college prep school before, and that will put her at a disadvantage, particularly coming into the program as a junior.
Most parents don't pay for college prep boarding school as an intentional prelude to an average public university. These programs are geared towards entrance into top schools. Not necessarily Ivy League, but there are a lot of top schools not in the Ivy League: Duke, Rice, MIT, Northwestern, Stanford, etc.
If the demands of the new school overwhelm her, her critical 11th & 12th grade grades might suffer, & that could make it harder for her to get into her university of choice. Think about the effect this could have on her future before moving her to try to make her more enthused about school. I'm not trying to rain on your parade, just asking you to use caution. Be sure to talk candidly with multiple admissions staff about her academic background and goals, and listen carefully to their opinions of whether or not she would be a good fit.
I have to agree with this. I am extremely bright (at least that's what the IQ tests say), and I transferred from a so-so school to a college prep day school (highly rated, but not up there with the prestigious boarding schools) in 10th grade, and I struggled. A lot. To give some perspective, my junior year, a previous graduate who went on to Harvard came back to visit. She told the 9th grade biology teacher that nothing at Harvard was half as hard as his class! Partly, I wasn't used to the intensity of the workload. And partly, I didn't have the right background. The school was a combined middle school and high school, and everyone else had taken two years of Latin in middle school, had a solid science background, etc. It was pretty humbling to go from being the smartest kid in the room to barely average. I rallied over time, did pretty well my junior year and very well my senior year, but it was tough. So make sure you find the right fit, ideally in a school that offers a lot of support for transfer students.
One thing I would give some thought to, before sending her to a boarding school, is that while all of the girls in our family are nice people, the girls who attended private school came from almost universally very wealthy homes, with expectations and attitudes that came from that lifestyle. They had similar friends they had been going to school with since kindergarten. I would want to make sure that I knew what kind of an environment my daughter was entering and how her personality might cope with being thrown in with kids who might, or might not, seem a little different and "clique-y". If she hasn't been to Europe several times, stayed in the very best hotels, worn the most expensive clothes, and taken for granted a million and one perks of money, she might find the pseudo-sophistication of the teens she could encounter a little hard to take.
Also this. My parents were both professionals, making a good income. But they came from working class backgrounds, so we didn't have the advantages of generational wealth. In the average public school system, my family would have been in the top 10%. At my school, we were poor. One of the annual fundraisers was a silent auction, and people donated things like a week's stay at a castle in Europe. Multiple kids took leaves of absence to go on safari for a month or similar pricey, long vacations. Everyone wore the very best designer clothes and shoes. One girl got a new Porsche for her 16th birthday. She totaled it in less than a week and got a brand new one the next day. After another girl graduated and enrolled at Vassar, her mom was furious that the airlines wouldn't let her prepay for four years of plane tickets for visits home.
It can also lead to some unexpected situations. I finally got somewhat popular my senior year and was invited to an 18th birthday party in a mansion. Little did I know that it was actually more of a business networking event. We were playing the stock market game at school at the time, and I happened to randomly be good at it. Which meant I got to spend the entire evening talking stocks with old men lol. Not that anything exciting was going on in any of the other rooms. The whole purpose was like a coming out party, only instead of debutantes being presented as marriageable, it was high school seniors being presented as the next generation of business leaders.
And my dad said the place was like a faucet you couldn't quite turn off, only he was dripping money instead of water. Every few days there was something else to pay for, whether it was a mandatory field trip or a costume for the school play or the week in Canada with French Club (not mandatory, but every single kid was going so it would have been weird for me to stay home). And it was a bit tough to make friends, because they had all grown up together, attended "feeder schools" since kindergarten, in many cases traveled the world together. I have no idea of your income or family circumstances, but unless you're generationally wealthy, prepare her for the idea of being the "poor kid," and talk through ways to adapt.