In many instances, the airlines (rather than the Transportation Safety Administration, which operates the security lines) have given people traveling with small children certain preferences in order to make their traveling more enjoyable. Still, whether it is the airlines of the TSA, you generally don’t have any travel-specific rights simply because you are child-laden. All of your “rights” are usually just voluntary accommodations that the airline, TSA or anyone else at the airport can take away at anytime.
The fact that your child was ill does not change anything either. Frustrating, I know, I travel with three small kids all the time.