I would willingly pay for limited access in the wee hours. I hope more museums will move in that direction. Whether such options are available in Rome, I don’t know. I was there for meetings and was able to stay over a day. I will be the first to admit that a day is not quite enough time to “do” Rome properly.
One of the best things I did was visit St Peters when it opens at 7am. The rest of touristic Rome is having breakfast. The hordes arrive later. There are small masses taking place in side chapels but you pretty much have the church to yourself or very few at least. You have to get up early of course but I got to view the morning sun streaming into the church from the front and that was also quite glorious.