The two things K often (though not always) has going for it are 1) it's often only a half-day program, so the child can be in a school setting part of the day and still have time and energy for his own projects the other part of the day, and 2) it's often focused on the social and the exploratory rather than the strictly academic.
#2 might sound contrary to what you want for a HG+ kid, but for a kid greatly advanced in academics, getting the social without being subjected to a lot of academic stuff that he already knows can work nicely. Our DS6 was very happy in half-day K with a good teacher who recognized and enjoyed his abilities, differentiated what work there was, but mostly let him do his own thing as much as she could. Then the other half of the day he pursued his more academic projects at home. He was very happy.
Our problematic year was 1st grade. YMMV, naturally, but since 1st grade is the big push-to-read year around here, it was the year DS6 had trouble. From what I've read, this is pretty common.
Check out what K and 1st grade are doing in your area. If your schools are like ours and K is play while 1st grade is the learn-to-read year, then I'd probably recommend doing K this coming year with plenty of at-home enrichment and skipping 1st grade the following year.
But follow your gut and the guidance of this tester, since she seems like a good guide for you and she knows the schools. If you think he needs more from school now, then do it!
