Oh, welcome to the club! I think you've gotten some good advice already and from your recent response sounds like you've solidified some of your thoughts, even if you don't think you're leaning a direction, so I just want to share a couple of things, if only to commiserate, as we're in a similar situation with DS5.
First off, I have to say I am always amused when people think that you can take an average kid and just teach them math 4 levels above their grade if you just *make* them.
We are considering (kind-of) the same 3 options for K, although we've almost entirely dismissed the public school as a viable option and the private school we're leaning toward is fortunately only 5 minutes from our house. I feel we're on the precipice of homeschool. Maybe not this year, but a good possibility for the next. He's currently in the PreK at the private and we're dealing with the same situation of very active boy not engaging in the work that he's given, talking constantly about "goofy" sometimes disruptive unrelated things. And I'm constantly at wits' end dealing with this stuff and can't imagine having him here all day.
Although it would be nice to be able to make school decisions based solely on academic fit, the issues of moving and commute are not insignificant. In reading what you wrote above, it sounds to me that you don't feel the private option is probably worth the other sacrifices. My husband and I have discussed the hypothetical scenario of moving and have almost entirely ruled it out because: 1) You don't know if it really will be a good fit, 2) The impact and implications on the rest of the family, especially siblings, including 3) leaving family and friends and base of support and 4) What's a good fit today may change in a few years, or 1 year, or 6 months. It's hard to tell with these kids.
Which leads me to the next thing: you want what you choose to be the best decision for now, but you also have to consider that it might not be the best decision for very long. That's why I'd rule out the options of moving towns and probably even buying a bigger house. I think it's a good idea to keep a framework and financial situation that's stable, but allows flexibility for changes down the road. For example, if he would go into 6th grade math next year and the school only goes through 8th, what happens when he's ready for more than 8th grade work?
Before I got to your most recent post I was going to suggest the option of tutors, either with partial public schooling or as an alternative to you doing the majority of the homeschooling yourself. This is something we're considering down the road. I have no idea if the au pair for all 3 is do-able; I'd lean toward having a couple of different tutors who are passionate about their subject and would enjoy teaching HG kids. I am reluctant about homeschool for all the reasons you mention, and one thing that we're going to try when we get to that point is not actually doing the schooling at "home". This may not always (or in actuality) be feasible, but as others mentioned, we would expect to do no more than 2 hrs of instruction per day, maybe not even 5 days/wk and we'd like to do that portion at the library, meeting tutors at the college, or at the school if we did partial enrollment. I think that would help preserve the sanctity (and cleanliness) of the home and help put everyone's brains in school mode with fewer distractions.
Not sure if any of that is helpful at all, but I know for me sometimes the action of telling/explaining the options for others helps me come to some clarity. Kind of like when DS does eeny-meeny-miney-mo to make a choice because he can't decide, but then ends up with the one he doesn't want and has to do it over until he gets the right answer.
