This is from an interview with a celebrity at her home when she is actually spending time with family. 73 questions with Kim (see:1:46-1:55)
At one point, where she is with her children and talking to the reporter simultaneously, one of her children coughs on her while she is trying to answer the questions of the reporter. So, when he coughs unexpectedly, she says:
Oh, cough on me, why don't you?
I did not quite understand what she meant by "....why don't you?", because this is used when you are making suggestions. However, it does not seem to sit well in here. Obviously, she wouldn't want him to cough on her more and more while she is trying to talk to the reporter.
Or is she saying so, because she is not happy with this coughing and is being sarcastic". But then it still would not sit well either, because when she said "why don't you?", the reporter is not laughing, on the contrary, he says "Aaaww" which shows he is feeling sorry for him being ill.
I could not be sure about the meaning of "....why don't you?" in this context. So I want to ask: why might she have said: "....why don't you?"