Cardinal sin and assurance that a battery will go to hell = NOT disconnecting battery when trailer is in storage. A battery disconnect switch is the easiest and best way of accomplishing this.
To test the output of the built-in battery converter/charger: When first powered by 120V and charging a depleted 12V battery, measure across the battery terminals. There should be 14V + present. After a short period of time, the converter will go into absorption stage and voltage should be 13.5 V (+or-) at the battery. Your new battery, fully charged, should be 12.6V, not 12V.
Walmart batteries will work but are at the bottom of the pecking order for deep cycle function. Two 6V Sam's club or Costco batteries will outshine them by a considerable margin. When connected in series, the 2-6V GC2 batteries can be charged with your original converter or any 12V battery charger.