OK, 1. This other child has been consistently winding your son up over a prolonged and constant amount of time. I think that would make anyone fed up and concerned about it potentially ramping up to a physical aspect sooner or later. Your DS reacted to what he felt was going to be the next (fairly logical) step on bullying pathway.
As minor as the wind ups may be, given the duration and consistency of it, I would have considered this to be bullying from another child, and as soon as I learnt they were sat next to each other again, I would have been down that school insisting they were to be separated.
As much as violence cannot be condoned, and alternative paths should be sought, it's easier to sit here as an emotionally mature adult and say that to a child, who is still going through the stages of learning and testing social boundaries.
As for the being rough and easily irritated at his brothers, and I note the mention of ASD and potential ADHD - as soon as I read the first bit, it sounded very much like my DD, who is level 2 ASD & ADHD. She struggles very much with her emotional regulation, and more often than not, will resort to physical means to sort out a challenging situation.
We sit, we talk, we have punishments and we explain and try to find coping mechanisms for her moving forward. There is a lot of repeating, and it is going to take a long time, but, she needs our help more than our punishment. Whenever she is punished, we explain to her that the punishment is a direct consequence of her actions. It is hard, and it hurts me when she hurts another child, because as a mum I would hate for another child to be scared to come to school because of her.
BUT... if said child had been long standing bullying her, and she reached her trigger point and reacted out of fear, I would partly understand. I wouldn't condone it, of course, but I would also be holding the teachers at the school partly responsible too. They hold a duty of care to all of the children there, and if there are pre existing issues with a couple of children, they need to keep an eye out and watch for triggers and signs of negative interactions.
Push through your GP for an ADHD diagnosis and possibly look at a school which is prepared and able to deal with your ASD DS. Perhaps he might thrive better in a smaller environment with better teacher: student ratio & provides more support / regulation centred learning.