My son has SPD with ADHD. It is my understanding that SPD is not a "Stand Alone Diagnosis" it typically is coupled with another neurological disorder. My son was blessed with ECI (Early Childhood Intervention) and our OT brought it up to us, we read "The Out of Sync Child" and "The Out of Sync Child Has Fun" to gain a stronger understanding of SPD. The official diagnosis came when we began seeing his Developmental Pedi just this past year - we were more focused on the initial diagnosis. We knew he had sensory issues, but we did not know what was the cause of everything else. The entire process from interventions to diagnosis was from 2.5 to 6. Now, part of the delays are that the doctors that can officially diagnose children with these types of afflictions often take 6-12 months to get into and more than one appointment to fully diagnose. Yes, that is 6months to wait for the initial visit and another 6 months for the next and typically by then you are closer to a diagnosis - but not always. It sounds to me like you have an accurate handle on the SPD issue, at least recognizing it's there - now you need to get a medical diagnosis.
Chances are school is not going to be able to speed the process along unless he is failing, at least that was my son's school's "party line" - then I got really pushy. It does not hurt to ask them to do an evaluation of your child, you can do so formally IN WRITING. They can not diagnose, but you may be able to find some help there - even if it is just a direction and some small interventions. Many of my brick walls with the school have been due to "official diagnosis" issues - since my son was young and they are also hesitant to run with too many labels too young.
All I can give you is my personal experiences - from your post I would agree with your direction of SPD. I am also going to say be prepared for further diagnosis, extended diagnosis time while determining all your son's afflictions and do YOUR research. I suggest you align yourself with a doctor who's school of thought/research etc. is proven, backed up by data and aligns with your own. Ex: for my family medication is not the first choice, interventions and positive support are. Our Developmental Pedi respects, understands and nurtures that while always giving us up to date information on medicines so we can make educated choices should the need arise. I like that he keeps us up to date on changes, new medications and research based on the medications he uses with out saying "Medicate Today." He also armed us with a positive rewards program that has changed our house! Together, we agreed to allow the school's 504/BIT plans to take effect, and work or fail - then revisit his needs mid school year. So far, my son is benefiting from interventions - this doctor is a win for us. We had a doctor that was a pill pusher and kicked us out of her office when we declined to heavily medicate our 4yr old - NOT for us.
Regarding meltdowns, they are a typical sign of an overstimulated child. When a child has so many sensory inputs, especially an SPD child, that they can not process them all at once they begin to shut down in a way as an attempt to shut it all out. Overstimulation is often the reason toddlers and babies get cranky and fussy right? Well, with SPD Kids, that never really goes away and their cranky and fussy is record breaking. One of the things my son's school does for him currently is taking him on frequent motor breaks. What this means for my son right now is a walk out of the classroom (5min total twice a day) to the gym where it is darkened (half lights on), quiet and bland and he is allowed to run it all out and create a blank slate of stimulation for himself. In that he also gets the adult one on one attention he craves and needs. There he can discuss any issues he may be having or successes he wants to share. He has two Special Ed teachers, the Principal and myself when I am on campus (I volunteer on campus) available to him for said breaks and he likes the variety. This WORKS for us. We also use these as a transition after a major event in school, Fire Drills, Half Day Field Trips, Visitor to the Classroom for an event - anything that might disrupt the classroom's typical routine. In addition, if he is afraid to try new things something that many SPD kids are afraid of, being "Teacher's Helper" may help here. So, being the Gym Teacher's clipboard holder and sticker passer IF he completes xyz activity might be a positive direction. IF you suggest this I am going to say that the first 5 times this is done the activity be something he CAN and WILL do to make the "helper" part a positive experience and something he wants to do. Once this is in place slowly add things he is less comfortable with and I do mean SLOWLY. Make it a do the 5 jumping jacks and try to jump the rope once. If he holds the jump rope that counts as trying. Step by step and the goal is that he will be able to participate in 25% of gym by the end of the year if now he is participating in 0%. Same things in art, music etc. Sometimes special one on one styled, teacher's helper incentives etc. are good tools. My son's classmates ALL understand that my son has a few extra needs and no one seems to mind that he goes on breaks, has a rocking chair for circle time/morning meeting and is "Teacher's Helper" more often than they are. I think they all see that the day is better when those things happen.
I hope, if you read all of this - I know posts like this can become overwhelming - you can take at least one helpful tidbit. I can only speak to my personal experiences with facts and full knowledge, and can only hope to help. This was and still is a good starter website for me ... maybe it too can help you.
http://www.sensory-processing-disorder.com/