Normally those abuse the terminals of a DPDT switch. The one side has mains on it, and disconnects (hopefully the line side) the mains, using one section of the DPDT switch to do so. At the same time the other half of the switch is used to connect the 12V output to the cooler side, but also, when switched off, uses the other connection of that side to switch the common to the 12V external input, so that it isolates the power supply output from the circuit.
So operation is mains switch one way and it is on, switch off mains and you can use the 12V input, but not both at the same time. Simplest fix is to decide which function you want, and go from there. Likely failure was the switch burning up from high current, especially as the manufacturers do tend to use a 6A rated switch for it, about the most common rating for a clip in DPDT switch, and cheaped out on the switch, using a less than honest one that is not really capable of actually handling 6A long term. the peltier coolers and fans tend to run to around 8A of current draw, a lot of load for a switch that was really meant for around 3A current, but, because it would handle 6A for a few minutes, was marked as 6A.
Fixes are to decide 120VAC only, put in a new DPST switch, and wire both line and neutral through it, and wire output direct to the peltier and fans. 12V only, just use a 10A switch, and wire through to the peltier (do not forget a fuse on the input, preferably with reverse polarity protection with a 6A 100V diode, though the only thing you will destroy are the fans) and ignore the 120VAC power supply, you can put it in an external case and connect to the fridge with a socket to match, which is common on some I see.
Otherwise a set of Schottky diodes, rated 30V 25A, ex PC power supply ones are both common, cheap, and come with a free heatsink, to allow either supply to power the fridge. No switch to burn, and you just loose 0.5V across the diodes, which means it takes a little longer to cool down the contents.
Most SMPS units are not going to complain about voltage being fed back to the input, though they will then draw current, as the on board diodes are biased off, but the regulation circuitry is still going to be drawing current, and long term will degrade the feedback optocoupler, as it will likely be driven on very hard, as the battery voltage rises above the set point, and the TL431 regulator turns on hard, thinking the output is high, and trying to reduce duty cycle of the turned off primary side down, so cooks the optocoupler LED. Diode on output solves that, low leakage current will just mean the PSU output sees less than 1V, which the resistor there to provide minimum load will handle comfortably.