If your battery won't hold a charge it needs to be replaced.
Your vehicle should charge your battery when the engine is running if it's wired properly. This normally doesn't provide much of a charge but it helps.
You MUST have a battery on the trailer if you are towing it on the road. If the trailer ever comes unhooked from the tow vehicle the on board battery will operate the brakes on the trailer.
Since you will be buying one you need a deep cycle battery, not a car battery; they can look similar.
If you always camp where you can plug in get a small cheap battery like a group 24. If you like to camp where you can't plug in and so must rely on your battery for power you probably should install a larger battery bank consisting of multiple batteries. You usage patterns and budget will determine the proper size.