I visited a Tut exhibit a few years back. It was mobbed during the day, but you could pay to get a limited access pass for 2:00 AM. There was reasonable number of people in the museum, and it was a lot better to be out in the wee hours than to deal with the crowds at noon.
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.