Emm. Is it too late for the private school? Many private schools accept kids after school starts.
Otherwise, did the school tell you that your daughter would be working with the really good special ed teacher? If they did, it might be helpful to remind them.
As for the sounding out of c-a-t, I would lose no time and go talk to the OG trained teacher (NOT the student teacher) immediately. Be honest and gently assertive. Ask them to test her. Ask them to test beyond grade level. Get the general idea of what will be on the test so you can prepare her for it ("Will she be reading and answering questions? Will she have to write stuff? etc.). I know that any test prep has a bad reputation around here, but it's critical to avoid surprises on a test. Little kids can shut down if things aren't right. If you claim she's gifted and then she doesn't do well on the test because something about it surprised her and scared her, you'll have trouble convincing them about her after that.
It seems so right to let people who seem to be experts run the show. This course often seems to be best, and being a pain is no fun. But at the same time, sometimes it's better to be assertive. A lot of teachers/administrators don't think past grade level, for example. It's been my experience that this idea just doesn't occur to them (not ALL of them, just MANY of them). Remember, HG+ kids are rare, and schools are focused on low performers.
Random anecdote: after my son had been offered a grade skip a few years ago (skip 3rd, start 4th in the fall), they gave him a 3rd grade math book to do. This was in the spring. They wanted him to work through the book so that he wouldn't be confused in 4th grade. It was trivial for him: he was already finishing 5th grade math and messing with algebra at that point. At a conference, his teacher told me, in a surprised tone, "He's doing very well; I actually think he might even be able to do stuff past the 3rd grade level!" I had spent the autumn in meetings asking if there was there anything they could do to help him work at his level in math. This teacher was at those meetings and had looked at the work he had done. Yet the message wasn't getting through.
Sorry to say that this problem has repeated itself a few times since then with other teachers.
So my advice is to push gently but hard and
get stuff in writing (at a public school or a private one). When you rely on a conversation, they may say something that you interpret the wrong way and vice-versa. Everyone sees the world through the lens of their own circumstances, and what seems so obvious to one party is not obvious to another. Example: everyone will tell you that Mrs. X is the best special ed teacher ever! You will make an assumption that she will teach your child, perhaps not knowing that she's oversubscribed or maybe even moving to Maine next year. From their perspective, they're just telling you about one great special ed teacher at a school that has three other special ed teachers. They aren't making a promise that she'll teach your child; they're just stating a fact about the school.
It's maddening to feel forced to drill down to a level of detail including, "And will she teach my child next year? Precisely what will she teach?" But sometimes you have to. My rambling advice is to pick the stuff that you're willing to leave up to fate and pick the stuff that's most important, and focus on the latter. Get details and get stuff in writing.
HTH.
