There is no need to eat breakfast. I never eat breakfast. I wake up pretty early (usually before 6), and I will only eat something around 10:30 - 11:00 am simply because by then I feel hungry.
The reason you (and I) cannot stomach breakfast is simply because during the night the digestive system goes on detox mode, and loads of toxins are emptied in the stomach/intestine to be dispatched in the morning. So when you get up, your digestive system is full of toxins. If you eat breakfast, it goes into absorption mode and reabsorbs all the toxins.
On waking up, the best thing to do is exercise on an empty stomach and drink lots of water (or herbal teas like peppermint or fennel), then after one hour (if you feel like it) eat your breakfast. Your body knows very well what it needs, there is no reason to force yourself to eat if you are not hungry (or to force yourself to eat if you are stuffed).
Best wishes,
Bernhard.