Hi Alan,
Borghese does not allow any bags in their exhibit rooms. You will be asked to "check it." For your personals, you can wear a money belt under your clothes to store your money, PP, cards, phone, etc.
Other than the above, your camera bag should be fine. "A small back pack" could mean different sizes to different people. Is it larger than 14"?
Vatican Museums and St. Peter's Basilica will allow a small to medium sized bag or back pack. I'd say no bigger than 14." But, no guard had a measuring tape, lol. If it's a bit over and not stacked full to interrupt viewing or hit something, it should be fine. It really is up to the guard at the entrance. I can speak from experience and what I observed during my visits.
You will go through security; get your things scanned. No big bags or backpacks allowed. No large (walking stick style) umbrellas. Water bottle was okay if put in bag or jacket pocket. No food allowed.
You can bring your camera into the Vatican, but the Sistine does not allow pictures.
St. Peter's will allow your camera but there is a special meditation chapel that totally restricts cameras and talking. There is a sign and the guard or two will remind people.
For the Vatican and St. Peter's, if you want to check a big back pack, there is a storage area on the ground level below the steps leading up to St. Peter's. I think the one in the Vatican is close to the security check in. I could be mistaken about the Vatican because I didn't check a bag.
The Colosseo does allow backpacks and cameras. If you're doing a Scavi Tour, check their website for correct info.
The Pantheon had no restrictions. I saw people in there with big backpacks & some with strollers.
Most other churches had travelers with their big backpacks and cameras. Visitors were occasionally observed by church staff.
eta The Vatican Tour I took allowed my tote/purse which measures about 14." Others had slightly larger backpacks that were allowed. But, thinking on this, it could be at the guard's discretion as well as the time of year. I went in December. Maybe during a much busier season, it could be different.
For St. Peter's, I had the same bag. I saw lots or people with larger bags or backpacks. It really depends. Below the terrace steps of St. Peter's on the ground level, there is an entrance to a checked bag area as well as restrooms.