So what are you asking? You say you have an electrical problem but don't state what the problem is.
First off, a battery is needed for towing, as this is what will engage the trailer brakes in case of a break away from the tow vehicle. So far as the camper, everything runs off of 12 volt DC except for microwave and air conditioner. You fridge will also run off of 110 AC, but when shore power is not present, it will also run off of propane (needs the 12 volt DC to run the controls for it, however).
When you plug into shore power, the converter (inverter? I always get those two mixed up) will not only provide 110, but will also provide 12 DC to recharge your battery. CAUTION - older converters do not have 3 stage charging, meaning they don't change to a trickle charge once they battery is charged. This can results in boiling your battery.