I don't know about beaches, but Salerno might be an option. It's a city of 135,000 that doesn't get many foreign tourists and has reasonable hotel prices. It would give you bus and ferry access to the Amalfi Coast without having to pay for lodging there.
It sounds as if the 365-day Campania ArteCard could be a significant money-saver for the parents if you plan to hit a bunch of the covered sights. I found it impossible to buy the card locally in February, so I downloaded the app and purchased my card there. That part was easy. It was a bit discombobulating when the app decided not to work for several hours one day. Fortunately, I had written down the 9-digit "Serial Code" when I bought the card, and I had no problem getting entry to covered sights using that code number--which is what I'd have needed to do if I didn't have data connectivity, my phone battery died or I lost the phone. (You cannot take a screenshot of the card displayed in the app to avoid the lack-of-connectivity issue.) Normally, you would open the app, go to My Passes, log in and show the smart code. It would be simpler to have a physical card, but I went to two places that were supposed to be selling them and had no luck, so I gave up. It's possible the more popular 3-day card is easily purchased--maybe even at some of the covered sights, but that's not what I needed.
Campania ArteCard website
The website is slow to load. You can buy the card on the website, but I didn't do that. I don't know whether the card would be mailed to you or you'd print a paper version yourself. I didn't try doing that because my philosophy is not to buy anything before I need to, in case the trip gets canceled or postponed.