I second the suggestion to post some menus. You mentioned in another post that you have been weight training but that you haven't lost any weight at all. What does a typical days' menu look like for you? How many grams of carb do you eat at a meal? In a day?
Even going all fat won't stop the liver from producing glucose; the body will just start using your own muscle to fuel the process instead of the protein you eat and you really don't want that.
One thing that helps in the Dawn Phenomenon (higher BG in the morning than the rest of the day) is to eat as soon as possible after you get up and not skip breakfast; the act of eating shuts down the glucose production in the liver that occurs during the night.