Runny nose: that happens. You totally lost me at "lost growth spurts".
I am astonished and shocked by the desperation of so many people on this thread, to keep the cats in the family even if they turn out to be the cause of this child's constant reactions.
Children do not exercise choice in this life, so all these adults who are choosing to have pets are NOT appropriate comparisons here. I also firmly believe it is NOT appropriate, fair or correct to balance the needs of pets alongside those of children, especially babies.
Human babies and children are vulnerable for many years, whereas an animal's childhood is short, and it must be treated as less vulnerable when making choices like this! Would you prefer to rehome your cats or your baby? (In answering this question, please consider why it is so much easier/less extreme to rehome a cat than a child!) Moreover, your cats would suffer a lot less from rehoming than your DS would, so I am again astonished that you are trying to make a balance here.
OP, you have a responsibility to protect and properly nurture your child, making his life as pain-free and healthy as possible. You may not be able to have an allergy test at this young age, so probably your only guides are exclusion and observation. However, those are pretty strong guides, and you yourself say you are "convinced" it is allergy.
I cannot believe you are still trying to have it all ways and make your son suffer.
Lost growth spurts, indeed. You are an idiot!