I agree, Captain Cooks is very close to the pool. Though it is technically in the Great Ceremonial House, there is an entrance directly into Captain Cooks from outside, so you don't have to go through the lobby or anything to get there. There is also some outdoor seating right outside the door to Captain Cooks.
I see it's you, DH and your babies. One of the adults can stay with the little ones and go get the food and bring it to poolside to eat. OR you can all go to Captain Cooks, where you can either eat inside or outside. We have eaten food from Captain Cooks all three of these ways.