Wow! It's been a long time since I've posted on this forum.
This group is actually one of the few that isn't dominated by Celiacs. A lot of them do (did) have Celiac symptoms, but didn't get the gold star diagnosis. But there are a few with neurological symptoms.

My son Tom has epilepsy. Not exactly tics, but to my way of thinking they're probably cousins. We went dairy-free first. It turned out to be his biggest seizure trigger and allowed us to get him off his seizure meds. (Major "Yay") Once off the meds, we were able to figure out that rice and coconut also triggered seizures. It took me awhile before I convinced myself to try gluten-free. Gluten doesn't appear to affect his seizures, but it does seem to play a role in his coordination. Once GF, he quit running like a toddler with his legs wide apart. He's better at kicking, throwing; he's showing much more interest in writing. Don't get me wrong, he is still delayed, but we've seen a lot of improvement with our dietary intervention.

Have you looked into DAN! doctors? DAN! is Defeat Autism Now! I know you didn't mention autism at all. Tom is not technically autistic, but he does have a few autistic tendencies. You don't have to be autistic to see a DAN! doctor. We've been seeing one for about a year. He ran a lot of bloodwork on Tom to determine vitamin / mineral levels (among other things) and we are supplementing accordingly. One of the many comments he made while going over results with us was on hypotonia. (And off the top of my head, I don't remember what it was... "XXX helps with hypotonia.") I've been really pleased with the progress we've seen with his help.

I'm assuming your son is in Special Ed? Or at least receiving OT services of some sort?

I think a lot of folks here found that they did have to give up dairy. As I wrote above, we gave it up first. I think you should definitely do both. And, I think you'll see some positive changes. I'd also recommend you give him a good multi-vitamin, if you're not already doing so.