I've never seen "trailers" for bikes on the Appian Way, but plenty of bikes! The road is very bumpy but you can get off and cruise therough the dirt or gravel paths. Many italians ride their bikes, especially out near Aqueduct Park. You're correct, Sunday is the best day to do so...
Some links for bike rentals in Appia
RS Guide link
Travel for kids
These folks DO have trailers.....
Rome by Bike
I'm sure if google bike rental on the Appian Way you'll see many other choices. Take some water with you, and a picnic. It's a great ride, though I have to admit I've primarily done it on a motorino...
Ciao,
Ron