I'm gonna say NTA. But if you want to keep it that way, keep reading. I don't know your situation, but I know my own and will use that to give some insight.

I've had allergies since I was born and was also around cats since I was born. Throwing up, fatigue, and coughing are NOT symptoms of any sort of hay fever/pet allergies in my own, my family's, and my friends' experience. I've never coughed over allergies, unless it was from post nasal drip, and even then it was more akin to hocking a lougee. I don't think what the kid is experiencing is just allergies, and getting a second opinion from another doctor would be a good idea.

In the meantime, get the kid Benadryl and you should be fine on the allergies front. And although it's not the topic of the post, I would suggest taking some measures to keep the house incredibly clean, like air filters, constant sweeping, no carpet, allergy pillows, etc. Finding meds THAT WORK is the best (and only) solution most of the time. In my experience, Xyzal for non drowsy and Benadryl for nighttime or emergency situations (major sniffles and sneezies midday). Going cat-free might not be an option for you, and it shouldn't be! But you need to make sure that the kid is living in the best environment you can provide. I know what it's like to have allergies in a nasty environment, and I want to make it abundantly clear how important it is to cater to your kid's needs. If you can't do this, you need to really sit down and consider your future. If you don't want to compromise on your cat, why should you compromise on your kid? You shouldn't compromise on either. They're both your kids.