I'm so sorry you are going through all this (and your little one as well!). From what I have heard, that age really is the most difficult for kids with sensory issues. I know for me (I had many sensory sensitivities as a child) my wost year was when I was 7. I was in trouble all the time and remember just feeling so angry and frustrated. By the time I was 9, somehow I had learned to regulate my sensory needs. I could control myself about a million times better - and I attribute that to neurological development considering I got no therapy or help.
Regarding diet, I am a firm believer in - can't hurt to try. Though I really think you need a control subject, usually husband or significant other, who doesn't know exactly when a trial is happening. Go on the diet for 3 weeks, then off, and have the OTHER person (can be teachers, etc) keep a daily log of their experience with your little one. I suggest this because sometimes the shifts can be subtle and I think its hard for us to evaluate clearly on our own. Though for us, we finally had to get allergy testing done to figure out our DS's food issues.
Finally, regarding an ASD diagnosis - I hear you! Our DS is only 3.5 but we went through a lot of questioning about that issue. We have since concluded that he's not on the spectrum. A big part of that was getting him evaluated by two experts who are specially trained to determine language disorders from ASD. They both feel very strongly that other issues (especially ADHD, sensory and language delays) can look very, very much like ASD because they cause so many similar symptoms. They turn to the traditional ASD criteria to help them - so, 1) is there obsessive behavior that interferes with "normal" activity (this includes stimming, severely restricted interests, etc) and 2) is there a severe communication delay (and this should include gesture, facial expression, emotional reciprocity, joint attention delays, just a expressive or receptive language isn't enough), and finally, is there a severe problem reading social cues. Just parts of those aren't enough, the all need to be there and they need to be "severe."
Obviously sensory issues and language delays can cause many of the same kinds of symptoms but things like eye contact and auditory sensitivity are NOT indicators of ASD. They can be caused by ASD, but they are really just red flags and can in fact be caused by sensory or auditory processing problems as well. For example, eye contact is also very low for kids with receptive language delays or hearing problems. If you can't understand what is being said to you, it is normal to avert your eyes. This is why people using those check lists on line to see if your child has ASD is so counter productive. They look for red flags that can be caused by many, many other things.
I know this is controversial, but after our own experience, I really believe that many kids diagnosed as being on the spectrum, aren't. They might have a lot going on, but I don't think they are in fact spectrum kids. Not that I think it matters much unless it is a non ASD kid in some crazy 40 hour a week strict ABA program or something that could be counter productive.
As with them, in your case I guess I would ask - would an ASD diagnosis change anything you are doing? If it would then I would go and get another evaluation. If the answer is no, then I personally don't think labels are that useful. Hope that helps at all!