Hi Kim, I took the Walks of Italy "Pristine Sistine" tour in October, and for me it was excellent, although quite tiring, with lots of walking. I can't really comment about how your kids might do with the tour. Certainly if you are four of a group of 12 or 15, it would be hard for the guide to focus on the kids.
It did mean a very early morning, in order to arrive at the indicated corner at about 7:15 AM. There was a small crowd at the museum entry until the doors opened, and then our group went quickly through the museum in order to get to the Sistine Chapel. It was wonderful to be in the Chapel with maybe 50 people present, as I have heard that it gets incredibly crowded and hot later in the day. Someone, not in our group, took a photo. A guard saw that and started screaming at that tourist, demanding not only that he not take photos but that he delete what he had taken. "You took two photos" she yelled, "delete them both!". The Sistine Chapel was a wonder, but quite overwhelming, and to appreciate it, I think it would be good to review the subject matter in the many paintings ahead of time. There is a little discussion just before entering, but no guide or talking inside the Chapel. After about 20 minutes in the Chapel, we went back through the Museum in some depth, then passed through the end of the Chapel again quickly, it was extremely crowded at that point, in order to go into St. Peter's. The guide took us around the inside of St. Peter's, then said farewell and we could stay in St. Peter's more if we wanted. I believe that photos were ok in St. Peter's and of course outside. The tour lasted until about 12 noon, longer than I had expected, and I spent some more time in the church, gift/book shops, and then the large square in front, before walking on to the Castello. It was a long and tiring morning with lots of walking.
The idea of a private guide who could respond to the needs and attention span of your family would be a great idea. I'm not sure if private groups can get in during that early morning time. You might even contact Walks of Italy to discuss this. They have an office and phone line in the US and I found their people on the phone to be very helpful and well-informed. I took other Walks of Italy tours in Rome, and they have various small discounts for Rick Steves followers, people who book multiple tours, etc. so be sure to ask about those.
Good luck, enjoy your trip, and don't let any in your family to get too exhausted to enjoy it all!