23 days with no recharge, or new ice is going to be a very tall order. Are you sure you can't plan your trip in a way to resupply with new ice, or to recharge your batteries every few days.

For example here's a typical battery powered cooler/freezer It will run for about 10 hours on 2 full batteries. Without anywhere to recharge, you'd need to bring something like 55 full batteries on your trip. A lot of the good roto molded coolers will easily keep ice for 5 days if you don't open them much. So as long as you could get ice 4-5 times along the trip that might be more practical.

You could also look into adding solar power to your setup to extend the run times, but I'm not sure how practical that would be on a white water rafting situation.