Battery or no battery????????

Battery.

You should have a battery. As vermilye points out, the breakaway brake system requires a battery, and in many states this is required equipment so a battery must be present.

Some converter designs, admittedly generally not ones that have been made too recently, require a battery to operate properly or to clean up their electrically noisy output.

If you ever want to stop quickly when traveling to use the bathroom or make and eat lunch or similar, you'll need a battery to power the water pump etc.

All that being said, you certainly don't need to have a big or expensive battery setup for the kind of camping you'll be doing.

Your lights are all 12V I'm not sure they will work w/o a battery even when hooked up. Plus you will eventually want to stop at a rest area or somesuch to make lunch or do something else.

And....if there ever is a power failure at the campground, it will be nice to have lights, etc. while the power is OFF.

Also some charger/converters don't operate correctly without a battery present.

If you have electric brakes, the trailer battery powers them when you trip the breakaway switch.

Your 3 way fridge won't operate without continuous 12 volt power.