Hi JenSMP - Welcome to the board! I hope you find it helpful. I'm not in your neck of the woods, so I've never heard of the school you mention. However, if you're school shopping for a gifted child, here are some ideas. It's great that you have options. FYI: the "best" schools are not necessarily the best for your child. Some of the schools in my area that boast the highest test scores, etc., feel their curriculum is already challenging enough for all their students, and make no efforts to differentiate, even when needed. Gifted schools vary in flexibility as well. General gifted magnet schools are great for most gifted kids, but there are different levels of giftedness and some kids will still need a faster pace. I believe the key to any school you find will be flexibility; make sure you ask prospective schools if they provide acceleration or differentiation if a student needs more than the standard curriculum, and ask if there might be any parents of gifted kids that you can talk to.
Just curious, why do you say the local public school is not an option? Some of the most flexible schools are local publics, who may not even have a gifted program in place. E.g., my local public school does not have any gifted programming until 3rd grade, but recognizes that my 5yo needs differentiation and is willing to provide it. They gave achievement testing to my DS last spring so they knew what level he was, and handpicked a teacher for him. (That said, he starts next week, so I don't really know how it will go, but we're optimistic considering all they've done for us so far.)
Homeschooling is also a great option for many gifted kids. Your child can learn at his own pace and you can handpick the curriculum. There are many homeschool groups that provide outings and classes. Check to see if there are any gifted homeschooling groups in your area.
The homeschooling parents here can probably provide you with a couple of links to find out what level your child is. Also, you might want to discuss this with prospective schools. You might be able to get him into 1st grade if that is where he belongs. It certainly sounds like some of the "behavior" issues might be caused by lack of challenge and inappropriate placement.
Those are just a few thoughts off the top of my head. Good luck!