Well I dont know, do you take any medication to help you to control your blood sugar? I would be interested to know why your doctor says that a Hb1ac below 6.5 puts you at risk of complications.
I have had consistent Hb1ac results below 5 for the past 2 years 4 months, before that they were never higher than 5.6.
I do not take any medication, I am diet and exercise controlled only. I follow the low carb life style, my diet has been checked by a state registered dietitian to make sure that I get all the nutrients that I need to stay healthy, my doctors are happy with what I am doing, nobody has ever said that I am putting myself at risk of complications or heart disease.
Well done you on getting such good results, but dont put your health at risk, if I were you I would ask for more explanation from your doctor, some links to the studies that he/she referred to.
All the best.