Hello and welcome!
We just got results back from our DS4 testing and are in a similar situation. I would recommend achievment testing like the others have mentioned. For us we are going to try our public school and meet with the superintendent to discuss a plan for DS. I am going to advocate for skipping K entirely and doing some subject acceleration. It sounds strange to me still...but I know he just wouldn't make it in a K class. I still know that 1st won't challenge him at all, but it would be slightly better and then he could do other work (either having it brought to him or going to another class). It's so hard to know what to do.
I have recently come to the conclusion though, that with kids that are HG/PG, that most schools (whether private or public) are going to need an individual plan for each student anyhow. No one-size-fits-all approach will work. So I guess it reaffirms to me that I don't have to do a private school because who is to say that they are more equipt to deal with a child like him. Chances are neither setting has met many ever so wherever they are they will need a special program. It's really about finding a school that will accept my child and try to meet his needs. It's sounds easy enough, doesn't it? : )