Ice works well as a cooling agent because it takes a lot of energy to get water to transition from the solid form to the liquid form. It is the energy required for the transition, not the actual temperature that keeps things cool. While dry ice starts out colder, it does not require as much energy to warm and transition to a gas. Dry ice works well for quick chilling or short durations, but not so well for longer durations.
Before investing a lot of money in food that will spoil, buy a chunk of dry ice and put it in the cooler to see how long it will last.
Before I had refrigeration on my boat, I learned to thoroughly pre-chill the icebox for several days before leaving on a cruise and to get the food and beverages as cold as I dare before loading the icebox with the food and beverages and fresh cold ice. A block of dry ice in the cooler overnight before leaving would probably do a good job chilling the box and then loading it with pre-chilled food and good ice. Keeping the cooler out of the sun will go a long way to preserving the ice.