How likely do you think it is that you'd be able to advocate for entry into first grade? Our ds had a similar lag between K cut-off and his birthdate, and we tried for early entry to K and weren't successful - it's extremely difficult to get either early entry or a grade skip where we're at. If you feel like entering 1st grade is the right thing to do for your ds, advocate for it! It can't hurt to try. If you're on the fence or if you try and can't get the skip, here are a few thoughts re staying in the grade he was "born into" for K/1:
1) K is often more about social skills (learning how to be in school, be with the group etc than it is about learning anything. For all that may sound like a total waste for a child who is capable of working ahead of grade level, it also might be tough to completely skip K and go right into first grade, where the rest of the class will have been through that first year of getting "used to the system". Sorry if that sounds bleak, really it wasn't for my kids - they had a relatively good time in kindergarten thanks to the social aspect, and had a tougher time with boredom later on in 1/2 grade.
2) If you put him into Kindergarten and a teacher gets to know him, you might be able to have that teacher support you in an effort to grade-skip from K to 2nd (rather than starting out in 1st). Same grade skip, just done at a different time. Even if you don't find a supportive teacher, the school won't be able to argue that he won't have appropriate ef skills etc when he's been in their building for a full year (unless he actually doesn't have the appropriate skills).
3) The test scores you have are all over the place - that could be maturity, it could be ADHD, it might be something else. You'll have a better hunch re what's up once he's actually been in school and you know whether or not there are challenges with classroom work etc. As the parent of two 2e kids, as counterintuitive as it may seem, I feel that having the kids *not* skipped during K/1 was important. If they'd been skipped ahead before we'd had a chance to thoroughly understand their challenges and how to remediate/accommodate in the classroom and on homework - they would have been miserable with a grade skip, and we probably would have set up a situation where a teacher could have easily said it's not working due to immaturity or the grade skip, rather than seeing it was a disability at play. That too might sound discouraging, but once we understood their challenges and once they'd had remediation and we knew how to accommodate, my kids were *not* held back by their challenges, they were able to subject accelerate as they moved on through school, and that's worked out well.
4) This is very individual according to child and their personality, but fwiw my ds hit a huge brick wall around the end of middle school where he was *bored* bored bored with school and frustrated beyond belief at not being able to learn at his own pace. In spite of that very bad time and that frustration over school, our ds has steadfastly maintained, year after year, that he is *very happy* that we didn't grade skip him - he's happier being with peers - even though he's on the older side of the age group.
None of those reasons I've listed are reasons *NOT* to do the grade skip - just a few thoughts in the event you *can't* get the grade skip.
Best wishes,
polarbear