What to do the last 3 days after the tour ends

You can never have enough time in Rome so I’d definitely add time there
Food tours are always fun-look at Eating Europe.com
Stay at Smeraldo

Borghese is not on your tour -one of Europe’s best museums-pre book timed entry
(Closed Monday)

Colosseum if if interests you to do the full tour you’ll need to book in advance

Any chance you can change your departure to Rome? Seems a pain to have to travel to Venice when you don’t really need more time there and staying near airport is a drag!

If you can change flight to fly home from Naples go there and see Pompeii as well as the Archeological Museum in Naples (closed Tuesday)

Unless you could change your itinerary and fly out of Naples, my suggestion would be to visit the ancient ruins of a Roman city in Ostia Antica instead of Pompeii. Ostia is easy to access from Rome city center. It takes about an hour to get there, and you can have a good visit in a couple hours. It is beautiful and has grassy and shady areas. You can see ruins of homes, restaurants, toilets, meeting places, mosaics in the market area, etc. There is a small museum.

You won't have seen "everything" in any of those cities. Perhaps, since you fly out of Venice, you'd want to head back there and see more of Venice? Or stop in Ravenna for frescos or Vicenza or Verona. Or stop in Bologna for a couple days?

Rome is definitely worth 2 or 3 extra nights. We spent 8 nights there and never ran out of things to do.

1. Yes, the Boss and I stayed in Rome after BOE breakfast Never can see everything in Rome. Would end the day with a food tour. Don't remember exact name but it was something like Food Tours of Europe (Rome and specifically Trastevere) Great guide provided background on food and area. I believe we walked to 6 or 7 stops. Included wine tasting, artichoke appetizers, pizza, etc. Finished at the only restaurant on the tour for excellent pasta. As I remember cost was about $100 each.
2. Consider private tour to Ostia Antica. About 20/30 miles west of Rome near airport. Was original Port for Rome before river became unpassable. Reachable by rail Our guide met us at our hotel and held our hands from the hotel to Ostia and back Unfortunately, I've lost the guide's card but the search engine on the Forum will help. Plan on about 45 minutes time to get to Ostia and about 3 or so hours for the tour. We went on a weekday and there were no large tours in sight
3. High speed rail to Venice and hotel near airport for restful night before flight home relieved our stress factor for security etc.

Have fun and happy travels

Consider day trips to Tivoli and Orvieto. And see if you can change flights and leave from Rome. Even if there’s a charge to do so, it might be cheaper than rail tickets to Venice and you’ll have a lot more time to enjoy Rome.

Found our Ostia Tour Guide Francesca Maria C Tours by Locals--Rome BA and MA in Archeology from University of Rome Born and raised in Rome Does private tours only Highly recommended

Which guidebooks did they provide to you as part of your tour package? I would take the time to look at them and see what appeals to you.

With only three days in Rome during the tour, you've barely scratched the surface.

Any chance you can fly home from Rome? Multi-city trips are often about the same cost at round trip. Talk to your airline and see if they'll let you switch. If there's a fee it might still be less money than train tickets to Venice. This will also save you some time you'd otherwise spend backtracking.

As you can imagine, Rick Steves has suggestions about what to see in Rome: https://www.ricksteves.com/europe/italy/rome-itinerary. Or you can look at the Best of Rome in 7 Days tour itinerary and see if anything there interests you. I'm nuts about ancient Rome but am trying to balance that with some Renaissance art and architecture when I make my second visit in October.

Yes, I've taken a Colosseum tour. Rick gives it three triangles, but if you wind up having too much to see, you can get the general idea walking around the outside.

I usually stay on in the tour hotel. This saves time: no packing and moving and checking in. Plus I'm already familiar with the nearby restaurants.
Have a great time,
Marty