Best way to tour Colosseum / Roman Forum / Vatican in a day and avoid crowds?

Be warned-we were in Vatican Museums on a Friday night in October and there were large crowds. I was also disappointed that the Pincotheca was closed. Hopefully someone who has done the early morning Vatican Museums tour will tell you if that's any better. There is also a light show of Forum at night. My husband said it was really good and the boys might like that better than daytime. However, getting teenagers up early when they are still on US time could be difficult.

Hope you have sturdy feet! That's a lot of walking in one day. I think your plan though basically will work for you. The only thing I would change is the guide at the Vatican. The Friday night opening has become my annual 'must do.' The hours are 7 - 11. You can buy a ticket that includes a very good cold supper buffet. At 8 o'clock there is a concert (mixture of Italian popular and semi-classical, at least when I've been there). This is all outdoors in a huge courtyard. You could skip both dinner and concert and just head into the museum. the RS book is fine - not sure a personal guide is necessary. It would appear that you finish the walk though the various rooms and hallways in the Sistine Chapel. You do. But at 11 o'clock the guards 'invite' you to exit. If you linger a bit the crowds (there aren't that many people) will go ahead and you can walk more slowly with the guards behind you. Surprise! There are still a couple of hallways to go through that have beautiful exhibits and (of course) souvenir stands. The windows are open, the fresh night air beats the humidity and excessive heat of the day, the lack of large numbers of people crowding you means . . . well, personal comfort.
I just re-read your itinerary. Again, I'm not big on guided tours (RS gives plenty of info). But I do think you'll want more time for the Coliseum, Forum and the Palatine Hill. It's a lot of walking, lots to see and you'll be happy for any shaded bench or rock to sit on. It's not just something you see, I think it's better if you think of it as something you experience (which means moving slower and reflecting more).
Have a wonderful trip!

Check hours that the Pantheon is open. It closed early while we were there over New Years week.

Near as I can tell, these are the current hours for the Pantheon.

http://www.polomusealelazio.beniculturali.it/index.php?it/232/pantheon

Visiting hours:
Monday-Saturday 8.30-19.30 (last entry 19.15);
Sunday and Sunday holidays 9.00am-6.00pm (last admission 5.45pm);
Non-Sunday holidays: 9.00-13.30 (last entry 13.15)

On the occasion of the liturgical celebrations, access to the monument is allowed only to the faithful.

Closing: January 1, December 25, except for extraordinary openings....

I have to say, that is quite a lot for one day. But, we did the Walks of Italy underground Colloseum, forum, PH. Crowds aren’t bad at all in the morning and I think you will really like the underground. With that tour, you get to go inside the Colloseum. It’s not open to the public in general to do that. Their guides are incredible and really bring those ruins to life. I would also start with that tour. I wasn’t exhausted like I was walking in the Vatican.

Please, do yourself and family a favor. After lunch, go back to your hotel a few hours, get off your feet, chill, etc. You’ll need your 2nd wind.

Based on our experience with the Colosseum night tour - I wouldn't flip it. In Aug 2019 my wife and did a Colosseum night tour, which included the underground. We were very disappointed. (As did a others on the tour: six attendees walked out early). I posted a review on the forum: https://community.ricksteves.com/travel-forum/italy/vip-colosseum-at-night-tour-by-the-roman-guy-a-great-disappointment

On the other hand, the morning of the same day we did an early morning tour of the Vatican Museum, Sistine Chapel, and St. Peter's. It was great: https://community.ricksteves.com/travel-forum/italy/early-morning-vatican-tour-with-sistine-chapel-with-through-eternity-tours-excellent

Thanks to all for the good feedback. Based on the advice, we are looking at keeping the proposed itinerary, making sure we rest up mid-day, and self guiding the Vatican night opening (adding the happy hour option).

So grateful to the kind folks in this community :)