Before posters come on bashing my point, I am the parent of an ASD 18yr old, the step parent of an ASD 26yr old and I have worked in several residential care settings with ASD children and am qualified as such, so I'm not speaking from an ignorant perspective. ( not an all knowing one either !)
ASD or not, getting his hair cut is a basic thing, like sitting at a table, eating with cutlery, getting dressed, walking calmly etc.
It is something he cannot avoid for the rest of his life, so it's something he's just going to have to get used to. I'm afraid leaving it longer between cuts is actually worse, as there's no 'routine' for him to get used to.
I'd also advise you to let go of the ''I don't like short haircuts'' idea, as it's more important what haircut will be easier and quicker to do for your son going forward. ( he's much more likely to be hurt by scissors than clippers for example)
I'd recommend picking a barber, have a chat with them so you can explain the issues your son has, and arrange an appointment for a quieter time.
I'd also recommend that it is your DH who takes him, this time, and every time afterwards, so he can start a 'routine'. E.g every 6 weeks, your DH, DC (and anyone else he'd need, but NOT you), your DH gets his hair cut, then your DS gets his hair cut.
I've said 'not you' going for the haircuts as you are probably already overly emotional about this subject, and therefore more likely to 'cave in' when he gets upset, which only makes it harder for everyone concerned.
For example, would get children arrive at residential school who wouldn't sit at a table. At the school, EVERY meal was eaten at a table. It took less that 2 days usually, with calm perseverance, repeatedly taking away their food and waiting for the meltdown to subside, then getting them back to the table to begin again, before they would sit at the table for the duration of the meal.
These were all non verbal, extreme end of ASD kids. It IS HARD, and the less emotional you can be, the quicker the change happens.
I wish you all the very best for you and your DC. x