Best of luck on your health journey!
I do have some pretty strong advice, and I say this as a former 300 pounder who no longer is — who at one time had a body that was a total mess.
Your trainer is working you way way way way way too hard. If you’re throwing up, its his or her fault, and you shouldn’t allow them to do that again.
Secondary, perhaps less helpful advice: Give up exercise. Permanently. I hated it. Still do. Instead I do movement. All day long now. Occasionally some of my movement even involves exercise equipment. In your case, I would probably spend the next 4–5 months stretching. That’s how I started. I wouldn’t throw up, but would inevitably pull something horribly and stop.
Then again, my advice is different than most. I would also tell you to eat whatever you want, whenever you want, in whatever quantities you want. Just slowly and deliberately, start changing your wants. That applies to movement as well.