Did you meet with the private school? I would at least meet with them, ask the questions that need to be asked, and then go from there. When I go and talk to principals I can usually tell within a few minutes whether they are sympathetic to the cause or not, or willing to work with you. Some principals will initially seem agreeable so you need to ask detailed questions about how things would be handled (or what options would be considered). So you could ask something like "Would you consider accelerating a child either a whole grade or for certain subjects, and if so, under what circumstances? Do they need certain test scores? How much would they be accelerated? If they are not accelerated (as in sent to a different class) but still advanced, then what would be done instead?"