I've never trained with a power meter, but I did do morning rides (4:30 am - 6:30 am 3-4 days/wk) for over a year, before my wife put her foot down. (I don't race, so I didn't have a good excuse for making her "a cycling widow" - her words. Plus, you know, crashes (3), because I couldn't see road hazards - a shallow construction trench across the road; debris in corners; etc. - at 20+ mph in the dark.)
Although my body did get used to riding early in the morning, I don't think I ever got to the point where my early morning rides were as fast as my weekend rides during the day. Rides later in the day often felt harder because, surprise, I was able to work harder. It wasn't easy to get my HR up that early in the morning, although eventually I was able to work up to doing lactate threshold sessions (using HR, not power).
Everyone is different, but IME it was difficult to make harder efforts early in the morning, and it took time to get to a place where I could sustain hard efforts at that time of day.