Due to cortisol amounts released in the body, would a person lose more weight if they rode a bike easy for 10 miles vs. riding a bike hard for 10 miles?

Very sincerely you should not worry about cortisol. Very low calorie diets and very heavy exercise over a sustained period of time when you're already lean and trying to get leaner are when you have to worry about that kind of stuff. In the meanwhile ... keep it simple.

Some might imagine that since either way you're riding 10 miles so the caloric burn would be the same but it is not. Riding hard for 10 miles means you're biking for a shorter period of time so you have to factor the number of calories burned is greater during that time plus the time you're not riding you're still burning calories plus you would be burning more calories during that time period due to having a longer recovery plus because your heart rate was elevated higher for a sustained period you would burn more calories after the fact due to EPOC. So the slightly caloric nod would go to hard for 10 miles but it probably would not be a large difference.

This is why trying to figure out how many calories you burn with exercise is for the most part a waste of time. A heart rate monitor is helpful but even then it is merely an aproximate measure.