Hi
First step is always to be sure you have adequate water in wet cell batteries. If that's what you have, check them first. Top them off with distilled water. If they are down a ways, do that and wait a day.
A lot depends on just how much
12V power you are using in the trailer. Indeed bad batteries can be the issue as well. With just the fridge, the fan should drop out. In some cases it's as simple as a shot thermostat on the fan. So, lots of possibilities.
First thing to do is to grab your multimeter (or go get one at Home Depot) and measure the voltage at the battery posts. You should be up above 13V with the converter running and above 12.5V with it not running. If you are down around 12.3V or less, that suggests that the converter and / or the batteries have issues.
Sorting bad battery out from bad converter is a bit of a tangle. First step is to see if the battery voltage (at the terminals) goes up when the converter is turned on. You can turn it on and off either by yanking shore power or by the breaker on the AC panel. Let things sit for about 15 minutes with power off and measure the battery. Then turn the converter back on. The voltage should go up by at least a few tenth's of a volt. More than that is ok.
If your 15 minute test shows the battery down past 12.5V *and* the converter seems to do something .... probably time for new batteries. Assuming you have a pair, they always get replaced together. Whatever deep cycle battery Costco has on sale this week is probably the most affordable option. Choices go on up from there to $900 a piece lithiums. How's the credit card doing
Bob