What amp hour is the second battery in the vehicle and camper?
How long will you be on the road, 1 week, 3 weeks etc?
Even though you're not staying in one place for any longer than a day, I would be asking how long will you be travelling between stops, anywhere from 30kms up to 300kms? It's the shorter legs that may cause a problem. Maybe a portable solar panel could be on the cards.
Quality ice boxes are great, keeps ice up to 6-7 days, but keep in mind if you are using it for drinks (beer, water, milk etc) you will be opening the ice box more than a few times a day which will lose the effectiveness it will keep ice by half. Maybe go the other way, drinks in the engel and frozen in the ice box, that way you only open ice box once or twice a day. But again, trying to find ice when you need it could be a pita.
if you get another fridge my guess is you will need another battery in the vehicle unless you can utilise the battery in the camper if its big enough and house the second fridge in the camper.
And with more batteries added, the alternator may struggle with out a battery management system
A cheaper alternative would be to invest in a shot gun and fishing rod and fresh every day