OP here. So we did stop overnight in Anzio. Here’s what we found:
This town doesn’t have lots of ancient architecture or fancy hotels. It does however have a fabulous beach and awesome restaurants. In fact, we had a magnificent — pretty elegant — lunch at Ristorante Turcotto, which is still owned by the family that founded it in 1816. Great food and awesome service.
We also had apertivi at sunset on the beach and it was one of the best sunsets we’ve ever seen.
The train is easy and convenient, and the Commonwealth cemetery is a short walk from the station.
Bottom line — Anzio is a nice break from the hustle and bustle of Rome, and worth a day or overnight stay.