TheFork users who ate in Capri have said really good things about Longano Ristorante Pizzeria, La Palette and Capri Kosher Restaurant. Maybe you should give them a try!
Some of the more popular Campano restaurants that offer gluten-free options on their menu in Capri according to TheFork users include:
Longano Ristorante Pizzeria, with a 8.7 rating
La Palette, with a 9.2 rating
La Terrazza di Lucullo, with a 9.3 rating