You have a very specific diet, and when you combine that with the desire for weight loss PLUS the extra rigors of running, you can get into a touchy area when trying to meet all three objectives. You might want to find a sports nutritionist who can look at your diet, your weight loss goals, and the demands of running to make sure that you are fueling yourself properly.
But to answer your question about what I do to keep hunger at bay - I eat a lot more than I usually do, and I increase my carbs through whole grains (brown sweet rice, farro, quinoa) and vegetables that provide the biggest bang for my buck (broccoli, spinach, bok choy, squashes, onions, garlic, etc.). I've always been a grazer, eating 5-6 times a day, so that doesn't change. When training I find that I sometimes get hungry just as I am going to bed, so I'll have a bowl of cereal. I don't sleep well if my stomach is growling.
Good luck. I hope you find what works for you. Training is the best time to experiment.
"God gave you your body as a gift, so you should take care of it." - My Mom