I don't have lots of trailer experience, but I've lived with DC refrigeration on sailboats for many years and installed and repaired many of them
Do you need refrigeration 24/7? Will you load the box before leaving home and then expect it to be cool all of the time from that point on? - If so, then there aren't many options except to have it run on 12 volts so it will have power while you're travelling
Top loading boxes make lots more sense than any conventional looking front loading box for the simple reason that battery wise, they are MUCH cheaper to run. When you open any front loading box, all of the cold stored inside 'falls out' and you refill the box with new, warm, moist air which your refrigeration system then spends the next hour or so (and lots of your battery power) to cool again. Since cold is heavier than warm, you can go into a top loading box and not dump out all the cold - Much 'cheaper' to run, if you're on battery power
Good 12 volt refrigeration all has one thing in common - It ain't cheap. Near the top of the heap are the Engel units (and a couple of other brands) which use a more efficient, less power hungry compressor to make the cold, but expect to invest $700 or so to get one
If you DON'T need your fridge to be cold 24/7, you have other options, like iceboxes and plug in units which you wouldn't need to run while travelling. Ice can quickly get expensive, but if you don't travel a lot or if it's only for short periods, the $700 that a DC cooler would cost you can buy a lot of ice
If you never 'rough camp' you could use a more conventional 120VAC fridge and run it off an inverter while you're travelling and then plug it into camp power when you arrive at your destination
Several variables - You need to select the method that works best for how you use your trailer
Don