Is the answer
Needles? Like in vaccinations?
A nightmare for kids
You can make them stop just by saying my name.
Some kids are incredibly scared of getting a vaccination shot
No matter your size, I will hug you.
Needles go into everyone and, uh, "hug" them, if you want to call it that. You could maybe use this as a euphemism to make it easier on the kid?
Discreet, people don't pay attention to me,
Needles are very thin and small, and with infants some doctors try to to do it in a way that the kid won't notice.
But without me, you can say bye to your pride.
Well, considering what kind of sicknesses you can get if you refuse vaccination, some have long-lasting effects which will make life very hard on you.
You can even say bye to your life.
You could die - in the long run - by refusing vaccinations