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.