Traveling with Teens/Tweens - Lake Como to Rome to Amalfi Coast

It is a long distance to travel, but you have planned it well (giving sufficient time for each stop). That said, there are some great coastal places in northern Italy too, so you could save all of transit that time and spend those days elsewhere.
You can easily take the train from Rome to Naples or Salerno and hire a driver from there (ask the company which location they prefer to pick up)--that will save you hundreds.
One hiccup--a day trip to Pompeii from the AC is very difficult. You could arrive very early and arrange for a stopover at Pompeii (driver with car so luggage will be secure).

I visited Pompeii easily while staying in Naples, so I haven't tried this myself, but I think this might work:

• Travel from Scala to the Vietri sul Mare-Amalfi station (which is in Vietri) and pick up a Metropolitano train to Pompei from there. Those trains seem to run every 30 minutes.
• Travel by Metropolitano train from Vietri sul Mare-Amalfi to Pompei. Travel time is 33 minutes. The Pompei stop is some distance from the excavations. There is train station near the excavations (Pompei Scavi), but it is served only by the trains on the secondary rail line running between Sorrento and Naples.
  
• Walk or take a taxi (or there might be a bus) from the Pompei station to the excavations.

Rick says in his guidebook that the walk from Pompei station to the archaeological area isn't very interesting, but it seems to me that would be preferable to starting the day in Amalfi and traveling across the Sorrentine Peninsula to a station where you can pick up a train to Pompeii Scavi. Alternatively, you could use each of the routes in one direction.

If you have extra time at the end of the day, the Salerno historic area (perhaps 1 mile from the train station) is fun to walk around.

My suggestion is similar to valedelphia's. Take the fast train from Rome to Naples in early morning., then the Circumvesuviana to Pompeii.Have the car service pick you up there and continue to the AC. It's a royal PITA getting to Pompeii once you are on the AC.

One issue about Pompeii from the AC/Positano is that it's much more expensive, and takes far longer than a daytrip from, say, Sorrento or Naples. (I am not addressing the issue that many travelers don't know that Sorrento is not "on" the Amalfi Coast.) I personally prefer Pompeii as a daytrip from Sorrento, without luggage. But the fact is that huge numbers of time-pressed visitors go to Pompeii with luggage, on the way to somewhere.

I have never read a complaint here about the free luggage storage at Pompeii. Heat and sun will be major factors in visiting Pompeii in July. Also, the entire area (not just "Pompeii) will be jammed with tourists, taxis, and car services. This area is a victim of its own success.

Are you aware that Ravello is at a much higher elevation than the coastline? No question, it is a highly desirable "view" and visit location, but it is ... isolated. (Bus to Amalfi) Check on the Ravello Music Festival dates, both to get scarce tickets and avoid movement problems on concert days. I'm not positive the kids will find Ravello interesting enough, but most posters here save their highest praise for Ravello.

You have an AirBnb reservation, so it's too late to consider doing Rome on the end of the trip, because of the need to travel back from the AC. Do you understand how many "seats" it takes to get from Rome to a town on the AC? The "fast" trains to Naples are much cheaper with advance purchase, but such tickets tend to be non-changeable, non-refundable. You don't want a slow train to Naples. And the train ends in Naples (or Salerno, for those staying nearer Salerno than Sorrento.)

https://community.ricksteves.com/travel-forum/italy/naples-to-sorrento-train-or-private-driver

https://community.ricksteves.com/travel-forum/italy/getting-around-amalfi-coast

I would go directly from Lake Como to Sorrento. It's a long day but your only really long day. We did that two years ago. We had a private service pick us up in Naples, and take us to Sorrento, (Aldo Limos).
Then , visit Pompeii, sleep a second night in Sorrento, then move on to the AC.
Keep all your Rome days together at the end.
Also, not sure that Lugano, Switzerland is worth it. Lots of high end shopping, and that is the Italian area, so not much different than your other Italy stops. It doesn't feel Swiss.
Are you aware of the extra crowding in Rome this year, due to the Pope's Jubilee, expecting millions more visitors to an already overcrowded city?
Amalfi Coast will be hot and packed with people. I'm sure you're aware, and will plan accordingly. As the transportation infrastructure south of Naples has been sadly neglected, leading to the congestion. There is no quick or easy mode for moving about.
You will love Lake Como- so beautiful and serene.
Pompeii is incredible.
You have a good amount of time for your family vacation. Make wonderful memories. Safe travels!