There are some states that use a later date like September or even a few using December. The reality is this particular child is not well-served to be on the young end of his grade. For some kids, especially precocious girls, it might actually be fine.
Grade level is one of those things that is uncomfortable to pull back later. It's also social. So a child with that many issues probably needs the extra time to bloom socially. He can be taught where he is, called K4, and then he can grade skip later if it's apparent that's a good fit for him. My ds has a gifted IQ with his fall b-day, and it was a better fit for him to be on the older end of his grade. In fact, we'll probably grade adjust him again at some point or do a gr13 or something. It has become very apparent that he clicks better with kids who are a bit younger than him and fits in those classes better. So for Sunday School, activities, anything, he does better if he's on the older end in a younger level class, rather than being on the younger end of an upper class.
None of that stops me from teaching him exactly where he is. You should work with him where he is. Having btdt on behaviors, I can sympathize that it's a challenge. I *didn't* get help on behavior when he was 3-5, and although I was teaching him it was horrifically hard. We brought in a behaviorist, into our home, began ABA, and that helped. You'll find therapies and people that have trite answers (like RDI and their oh we'll just do this and he'll fall right in with a mentor/apprenticeship role!). Fine, whatever. For us it was lots of layers of things. Working on behavior means less academics, and then you'll feel guilty about that. The WORST combination possible is not working on academics AND not having effective behavioral interventions. That's the one thing to avoid.
Evals will save you time and grief in the long-run, because they'll allow you to cut to the chase and find effective behavioral interventions. If standard parenting would have done it, you'd have done it by now. Most kids, typical kids, are working on obedience from ages 2-5 and by 5 have it pretty well nailed. They have their sense of group plan, who's in charge, what their place is, and are by age 5 transitioning to work on self-regulation and self-control. So, when I worked in K5, the kids started the year with teacher control and ended the year with self-control. That is the BIG GOAL of K5.
There is a curriculum, btw, that they use with this age. It's something you could look into or something a behaviorist could do with him if you brought one in. It's We Thinkers by Michelle Garcia Winner of SocialThinking.com This is the stuff behaviorist, SLPs, etc. use with kids to help them begin to understand concepts like group plan, body in space, etc. It uses picture books and activities and is EXCELLENT. You can also find social skills groups sometimes that are doing it. In the groups my ds attends, there are some kids who have ADHD or no diagnosis yet, and there are kids on the spectrum. The kids on the spectrum need *more* intervention and instruction to get there and the kids with ADHD catch on faster. So my ds has worked through a lot of this curriculum 1:1 with a behaviorist, and now he's able to do it in groups.
Are you still with this ped? He seems really outdated on his understanding of developmental issues. Our ped had me going to the SLP for my ds' apraxia well before he was 2. Waiting till 3 is way late, depending on what you're seeing. And to suspicion autism and not do a referral is, I don't know. You're saying pretty significant things, things that ought to get you referrals. My ds wasn't diagnosed til 6.5, and I look back and go wow, he was clearly diagnosable at 2. That's a lot of time where we didn't have help we could have had.