We have eaten there at lunch with our 6 yr old dd and you can pretty much request any items that are on the kid's menu at the other Dolphin restaurants (the basic kid's stuff).
They are very accomodating. I believe the grilled cheese we asked for was even made at FRESH but I think the fries came from the Dolphin fountain.
Even when we ate at PALIO at the Swan she wanted a grilled cheese and it's not on their menu but the server (he was just great!) was more than happy to "fetch" one from Gulliver's.
The S/D restaurants are very good about that.
So...YES Children's menu items are available-just ask your server...FRESH is a MUST DO on our list EVERYTIME we go to WDW now!