Small child?
I am not sure how the small child fits into this scenario?
If a pilot (any pilot) is fatigued for whatever reason then it is incumbent upon that pilot NOT TO REPORT to fly; may not be popular with management/rostering but necessary.
All (or most) of us have had a small child in our lives (some several) at some time during our flying careers and yes! sometimes they yell and cry all night because they are unwell or whatever, but hopefully we work around it. We (that is the sexist men) pull on the earmuffs and the little lady (wife) takes care of the child child's needs.
My question is why was said pilot up all night with a young child when she knew that she had heavy flying schedule the next day? Having been up all night why did she not call rostering and inform them that she was unfit to fly due to fatigue?
Whatever the reasons(s) for the above this situation should never had happened in the first place AND when the Captain concerned realised it was begining to get out of hand quite correctly that Captain informed the company that she would not exercise Captain's discretion and that is HER prerogative.
The danger here is that the company may decide that this employee is a liablity and use it's prerogative to let that employee go. However not on the pretext of that employee not excercising her discretion but because a company cannot run a business profitably if aircraft end up at the wrong airport and the pax are pixxed off!