There are many ways to do what you want, separate systems is one way.
In truth do some searching around here, there are many threads, but adding insulation
is always a good thing, making it smaller also cuts down on requirements.
I went with a Cool Blue, but there are many out there thankfully, some water
cooled, some keel
cooled, some air cooled. Each has its proponents, then there are of course the portable fridge / freezers, Engel is seemingly the favorite, but expensive, and there are drop ins where you remove your entire box and drop in a self contained replacement.
Good news is you have many options, bad news is you need to pick one.