It's an amazing present to give, unfortunately you aren't likely to get the result you want. Good Italian food is made from fresh ingredients and eaten fresh, anything cooked and then frozen or refrigerated before being reheated will lose a lot of flavor and texture. Plus, part of the overall experience is going to the restaurant itself, the service and the decor, choosing from a menu, etc. I don't know personally but I've heard that there's a few good Italian restaurants in Pittsburgh, maybe the present would be taking your daughter to one.
If you want to try this it is workable, provided you have the right equipment. You'd need a real cooler, not a cool bag as those don't insulate well enough, and lots of freeze packs. There are coolers with built-in mini-fridges that run off a car's auxiliary power port (cigarette lighter) as well. Both solutions require the food to be refrigerator or freezer temperature when put into the cooler (cooler mini fridges can keep things cool but don't have the power to cool hot food down fast enough), and if you are using freezer packs they would need to be fresh as well to make it 10 hours. This will be a logistical challenge, if your relative is driving from Florida feeezer ice packs will be several hours old before the food is picked up.