You get the "put her lady's health second" thing because we(reader) know mao mao's father. she doesn't. (I heard it called god eyes view thing)Quite immature of the maid to think about the status of the doctor first, instead of her consort's health. If he is the most skilled man to do the job, that should be a good enough reason to bring him.
Quick edit since people seem to misunderstand what I'm saying: this isn't a babysitter job, they are looking for a doctor that would do an operation. If the only surgeon that could do a life-saving operation on me committed tax fraud, I'd 100% hire him. The maid doesn't even ask what kind of crime it is, which shows she puts her lady's health second to her reputation, which is wrong.
Even mao mao got scolded for not explaining thing properly. she went straight to "We gotta hire this skilled but criminal docter, this is only our choice. trust me,ma'am"
She has bad habit about jumping the gun too fast which sometimes good at resolve emergency but most of the time, it make people misunderstood her.
If only she laid out info about him being her foster father who teached her all medical skill(which help them in inner palace many times) the reaction might be softer than this.