Hello,
my little boy has some problems along the same line. He was diagnosed with central apnoea, and also with periodic limb movement disorder, when he was 1 year old via a sleep study. They said that the apnoeas were happening for longer and more frequently than normal, but as it was central and not obstructive at that stage they wanted to wait and see if he grew out of it. He didn't, and it got worse, as the severity of his allergies and asthma got worse with time. He developed obstructive apnoea. About 6 months ago his paediatrician sent him to see an ENT, who found that his adenoids were huge, and his turbinates were so inflamed from the allergies that he had no breathing space at all. He removed the adenoids and cauterised the turbinates, and it's made a big difference. I can hear that he's breathing better at night, even during the day, and his nose is drier than it used to be (I lovingly referred to him as my little snotmonster prior to surgery lol).
I see that you have already been to see a sleep specialist - seeing an ENT would be another option, but if his adenoids and tonsils are not that enlarged then an ENT might not really say anything different anyway. Maybe a paediatrician would be a good option for you? That way you could sort of look at the whole picture and work towards a definitive diagnosis. One thing that sort of comes to mind is that it sounds like something my other boy used to do, and it turned out he had reflux. He didn't seem to have the pain that you'd expect with reflux, he'd just stop breathing and then he'd start again with a big gasp. I was surprised by the diagnosis but when we treated it the problem stopped. There were no other symptoms, which made it difficult to diagnose, but once we got there it was an easy fix.
If you're sure it's apnoea, then maybe going back to the sleep specialist (or a different sleep specialist) and asking specifically for a sleep study might be a good idea. They measure his brainwaves and oxygen levels and so forth, so they can see exactly what's happening when he has these episodes.