We went to CA Grill with our entire family, my parents, brothers, sister in laws, my husband and children ages 11, 8, 5 and 3 and the kids did really well. Of the signature restaurants I've been to, so far CA Grill is the most kid friendly. We went late, so the kids could see Wishes from the rooftop, which turned out to be a great mid dinner diversion. It's also nice that the Wishes music plays in the dining room.
I would note that the two times we've been to CA Grill there's been a bit of a wait and that when we were there with the little ones, my brother, sister-in-law and the 5 and 3 year old took off after the kids were finished with dessert. The CM will serve the kids at whatever speed you ask them to, so if the kids are hungry, they can be served all of their courses first which also means that they will be finished well in advance of the adults.
On our last visit to the World, our kids were older, but there were LOTS of smaller children in the dining room. I think it's worth a try! And I agree with the PP who mentioned that you can always get your meal to go if there's a meltdown...at least it will be absolutely delicious food in a box.