I hate to ask, but I'll admit that it makes a difference to me: was the lady white or black (or other)?
What would I have done? I'm almost positive my reaction would've included a forced smile and these words: thanks, I got it. I would like to show even more appreciation and kindness, but in the heat of that moment you're experiencing frustration, anger, embarrassment, exasperation... you just want it to be over. Any input from anyone else just compounds those feelings, even though you appreciate their efforts. Plus, no matter what, nobody knows your kids like YOU do. Things may appear to be one way to outsiders, but they don't really know what's going down. Now my sister?? My sister would go off. She doesn't play when it comes to her children. She would really have hurt that lady in Target.
I witnessed something similar on the train a few months ago. This lady would get on in the evenings with her 4 or 5 year old son. This boy was B-A-D-BAD. I think he may have had some sort of developmental issue or perhaps ADHD or something. But he was a handful and she struggled with him every single day. This one day, he screamed and cried and yelled and fussed from the time they got on the platform and continued once we all boarded the train. He was calling his mom names and swatting at her. She was just worn out. Had gotten to the point where she wasn't even trying to reason with him anymore. She had tears in her eyes and a look that said "If I can just make it home...." I felt so bad for her, but I don't mess with people's children unless they are physically touching me or about to hurt themselves.
Anyway, this boy had the train lit up, so this lady turns around and looks him eyeball to eyeball - she was about 2 inches from his face - and said barely above a whisper, "look here, nobody is in the mood for all this noise. Most of us done worked all day and right now, we just want some peace and quiet, so hush that noise up. NOW!!!!! Do you understand me?? You're NOT getting a cookie. So hush it!!!" The boy made a grunt type of sound and hit his mom on her lap. The bystander lady grabbed him by his arms and went off on him... she was telling him that he better not ever lay his hands on his mother again or any other woman for that matter, and that if he made another sound she was going to take her belt off. She told him to sit his behind down in a seat and not to move until his mama said it was time to go. "And don't ask for a cookie or nothing else today, either!!!" LOL! And the lady wasn't crazy or anything at all... she was a regular professional, working woman, well dressed and nicely groomed, who was really frustrated by the way this boy was acting.