Sometimes rigid thinking is a symptom of OCD (obsessive compulsive disorder) - though you would need to see a mental health professional to ascertain whether anything is amiss. Sometimes, perfectionism is just a trend in personality and not part of a mental health diagnosis. In either case, it is important not to have people in the family give in to your daughter's bullying. There is no need to do everything her way. Let her deal with her anxiety herself and this will actually help her become less anxiously perfectionistic. Bach Flower Therapy can sometimes ease the symptoms of perfectionism when they are moderate or mild.  If her behavior doesn't improve after these strategies you might talk to your doctor for further direction.
Sarah Levine