Oh god, you have taken me back to those horrific nights when I thought I'd actually either die from lack of sleep or become completely deranged from it.
Mine would sleep for 20-45 minutes a stretch. And no matter how long he slept for, he seemed to manage to wake me at the worst point in my sleep cycle, so I was discombobulated.
He was late to talk too, which didn't help with him being able to tell me what the problem was.
We tried all of the methods and like yours, he'd just end up vomiting.
As soon as he was old enough to talk he told me me about baby jail. Basically he hated the sides on the cot. Not that he spent any significant time in the damn cot.
He is 9 now and can tell me how much he hated the cot. He also runs really warm in bed. He has a 4tog duvet and even that's too warm for him, even in the depth of winter. He now has a selection of sheets and blankets to help him get the temperature right.
With hindsight, I wish I'd have slept way more at other times. Weekends, early evening, whenever there was another adult about to have the baby/toddler. Because tackling anything when you feel like you do when your sleep is shot is almost impossible. Even smiling was a monumental effort in those days. I had to have a photo taken for a new driving license. I look like a ghost.
I also learned that there wasn't much that would change the situation, but perhaps I'd have taken the cot sides off much earlier, tried lightweight blankets, and shit like white noise earlier.
But, most of all, I should have talked to my friends and family. If I now know there's a non sleeping baby in the family, I offer to have that baby over night and accept that I won't get any sleep for one night while the parents get a good solid sleep. Have you got anybody like that who'd have the baby from say 6pm to 10am. If I did it on a Friday, I'd just sleep the next day, something you guys don't have the luxury of.
I totally feel your pain. Hoping something gives sooner rather than later.