I think if I was in your shoes, I would set up a time to tour your local public school and to see what it is like/what kind of services (beyond the pullout) are available. That's assuming you have not already been there. FWIW, it's probably wise not to lead with too much "gifted" wording until you know their philosophy (and how to word your questions/comments in line with that -- it's sort of ridiculous, but unfortunately necessary. Consider yourself a spy of sorts until you know what you're dealing with...). You mentioned your daughter is already 8 -- so perhaps acceleration in the public school might be an option. (There are a lot of 8 year olds in elementary schools around here, so in some areas, she would not be too far off age wise, which I think matters to you.)
We have been at three private schools. The first was an excellent Montessori, but we moved out of the area. The Montessori schools here are weak for elementary students (at least advanced ones). We tried an "exclusive" private school which turned out to be quite rigid and downright mean spirited (the administrator, at least). Currently are at a parochial school that is extremely flexible (they have offered several acceleration options!). Our child is 8 and in 3rd grade (no accelerations, summer birthday and started at 5). Accelerated about half the day by two years. This is the latest variation and it seems to be working well. He still finishes early (in the two grade up classes) but is happier.
