I don't think you should eat when not hungry. If you're consuming less calories, but still having the same amount of energy, that's because you're more adapted to ketones. I think if you were to eat more meat you would produce more glucose, and more ketones at the same time, but you would just urinate the ketones which are now surplus and run more off of the glucose. Eating anything with carbs would for that time take you out of gluconeogenesis, and therefore halt ketone production, and if you were to increase fat beyond the point where you feel full then you'd probably have intestinal discomfort.
Sounds like you're good as it is.
Even aside from whether you need more food, you need to be hungry to actually digest the food properly anyway. So even if you did need more food, you should look at it, smell it and think about it until you were hungry anyway, before attempting to eat it.
As for why running may contribute:
If you're running you're going to be using more protein for repairs. This would mean that your body would have less of it available. Ketones are produced anyway when your body converts protein to glucose, so if your body has less protein available for conversion, then it will want to make use of more of the ketones, rather than eliminating them.