Hi,
Sounds like we have some similarities. My DS started school last year, and had problems similar to those you describe. It was the first real clue that he had any SN, like your DS he hit every milestone in his development.
DS's school took the opposite approach to yours: they sent him home on a reduced timetable as soon as it became apparent he needed extra support, and kept him there until it looked as though they had done enough to give up, and they essentially expelled him at the age of five. Right now we are in limbo, DS receiving an hour a day of home tuition while we wait to see if we can get him a place at a special school.
Like your DS, mine is intelligent and is fine with eye contact and physical contact and makes friends to a point. I was worried about the stigma and the label a SN would bring, but someone pointed out to me that although DS has the intelligence to sit an exam, without the right support, we would be lucky if he would cope with entering an exam hall.
Obviously, the position you are in is just as untenable as mine was. You cannot continue to act as an unpaid TA for your son for the rest of his school life. What if you want/need to get a job, or you have another child? School need to be putting in extra provisions, whether or not they can "afford" it. Is your son on School Action Plus? If so, school have funding for him. If they need more funding, they need to be gathering evidence and making an application for a Statement.
What they are doing right now is the cheap option. It's also the option that will leave you with nothing if they decide to pull the rug out from under you at some point. And this, unfortunately, is a very real possibility. Have a read around on this board, you will see how often it happens. It's been a real eye-opener for me.
My DS had a diagnosis, a statement in the works, full time 1:1, had reports from every professional going and school tried to send him to a centre for badly behaved kids despite the fact that he has autism. The head promised us that he would ensure that DS could not get a place at another mainstream school anywhere local.
There was nobody who said to the school "that's discrimination, he has autism, his behaviour will never improve without the right support".
If your school doesn't want to, they won't adapt for him. If you don't have something screwed down in writing with a statement, your DS is entitled to nothing.
Actively seeking a diagnosis is a very personal decision. In my case, it was a useful one, both psychologically and in getting DS's case taken seriously. When we decided special school was the place to go, being able to say "he has autism" was useful. It also opened up a world of support outside school too. You can get a statement, DLA, and lots of support without a diagnosis of course, so it really is something you have to come to on your own.
Sorry if this sounds very bitter and cynical, but I'm a year on from where you are now and that is unfortunately how I feel. I really did think school wanted to help DS and had his educational interests at heart, but now I firmly believe they intended to expel him from the start. I'm so glad we pushed for every professional report, every intervention and every step on the path to a statement. It's meant we have a mountain of evidence to show our DS has SN and needs support wherever he ends up next.