I am not a physician, but the mom of a diabetic. I would suggest you check your sugar at that time of day. To see what your sugar is. I would also discuss this with your doctor. You could just be getting a low blood sugar in the middle of the night.
Good luck.
Even non-diabetics have blood sugar fluctuations and that might explain your cravings. On the other hand, many of us develop habits that become our normal ways of eating. It can be really hard to tell if crave a certain food because we're accustomed to having it, or if our body physically is in need of soemthing.
Sweets tend to give a quick rise and then a blood sugar plummet. Other foods can give folks (even non-diabetics) a steadier blood sugar because they don't cause a surge of insulin to respond to the sweets. You'll read in many sources that sweets are fine as treats and that our bodies will, long term, be healthier if we learn to enjoy more "complex carbohydrates" that take a bit longer to metabolize and thus keep us feeling satisfied for longer.
If you are interested in experimenting, you might record your pattern of eating sweets. You've already noted one pattern (eating sweets in the morning). Try varying what you actually do eat to see if, within a week, that craving doesn't go away.
None of us are physicians here, so it's good you're getting checked by one. Given that diabetes is in your family, everythign you can do to keep your good weight and to eat well will be good in the long run.