I don't think they're picky eaters...... when they eat traditional ishgardian meals though, they seem very picky because they're used to ennius making them food (and aymeric adopting other cuisine when it's his turn to cook). When they visit Darcy in Shirogane , they love that food too!!

Its really only ishgardian food they are really against, and of course personal dislikes

Amelia doesn't really like... slimy food. She'll eat it but it's not her first choice! (Ie: oatmeal, grains. It needs to be hard!)

Faust doesn't like certain veggies, but he'll eat them when they're fried or you don't tell him it's in the dish.