I suspect that his sleep might be linked to whatever is causing the vomiting. My mummy gut is saying that something is going on.

I know that in private sleep schools your child is admitted to the hospital under the care of a paediatrician. When I went the paed did an initial check on admission and was available every other day for a check up if we wanted. Might be a good way to tackle the sleep and health issues at the same time?

It sounds yuk. Hope you get it sorted soon.