Insulated lunch bags do not need to be refrigerated when you have the food in them as long as you plan to eat the food within 4 hours.
You should also add some cold ice packs that you freeze overnight and then put them in the insulated lunch bag to help keep the food cold in the lunch bag.
If you are not gonna eat the food within 4 hours then you need to refrigerate the food and lunch bag to keep the food cold enough to prevent spoilage.
However most people who use insulated lunch bags eat the food well before 4 hours so you should not need to refrigerate the insulated lunch bag.
Just make sure to use ice packs made for lunch bags or lunch boxes and the food will remain cold enough until you are ready to eat the food.