If you drive less than 2 hours and always have electrical hookup at the campsite: Then pretty much any cheap 12 volt battery will work. Especially if your refrigerator has the option to run on propane. If the refrigerator has a propane option your battery is pretty much just there to power the trailer brakes if the trailer becomes detached from the hitch. True the refrigerator running in gas mode draws a small amount of 12v power. But a standard automotive starting battery will run a propane RV refrigerator for at least a week.
If you go to developed campsites but maybe do not always have electricity for the weekend then you will want 2 batteries. Yes lithium would be nice. But you would need to figure out if the current built in charger has a lithium setting. If not, I would go with two 6 volt batteries instead of two 12 volt batteries.
Just bring an old automotive starting battery, throw it on the tongue and hook it up to power the breaks in an emergency. Tow the trailer home and then worry about what battery or batteries you need.