Although it would have been rude to just bring the sick child, at least she asked. What is wrong with asking?
Because it puts the other person on the spot to say yes. And it makes them wonder about your ideas of sickness and health. Honestly if someone asked me if they could bring their kid with chicken pox to a party I would have to wonder if they realized how serious a disease it could be, how uncomfortable it is for the person who has it and how communicable it is. And then I would start wondering what else they don't take seriously that they should.
Most people seem to think that chicken pox is only a mild childhood illness with no serious repercussions for many. And it is. But if you happen to be that one person, who for some reason have some sort of defective immunity, it becomes a life threatening illness. If you are an adult, it DOES become life threatening (and I'm not talking about shingles, that's a whole 'nuther story).
Right before the chicken pox vaccine came out, the lab's post doc was exposed to chicken pox through her nephew. This was a woman who worked with microbes all day long without issue. But catching chicken pox put her in ICU for 3 weeks and she was out of work for nearly 6 weeks. The scary part of this was that she had had chicken pox as a child....she thought she was immune
OP, you did the right thing. Don't let her attitude get to you.