I am a type 2, but I struggle with high morning numbers as well. I don't consider my morning numbers my "fasting" level. My fasting level is the level 3-5 hours after a meal. My morning numbers are my Darn Phenomenon (DP) numbers and are basically disconnected from my fasting numbers. DP happens to a lot of us, different people have different explanations, but in the end our livers dump glucose in the morning which cause a sometimes disturbing rise in blood sugar. You can fight this by raising your overnight basal levels, but for many of us you can only go so far, when basal levels go too high we have hypos during the day.
My approach to DP has been to establish a basal rate (about 24 units/day) that keeps me right on target when fasting between meals and when I sleep. I split my basal injections, so I take a morning shot of about 12 units. At night, I take the rest of my basal, but I also stack up more insulin to help my overnight numbers. And I stack about 3 times my basal rate, attempting to time my evening injection to peak in the early morning hours of 3-5am. Despite this, I still often struggle with higher morning numbers. So I just do my best, and accept that if I awake high I will have to correct.