If you want to run lights in your horse trailer using a battery that charges while the trailer is hooked up, it's generally a good idea to use a charge controller to regulate the charging process and ensure safe and efficient charging of the battery.

Instead of using the "neutral wire," you'll need to connect the positive wire from the trailer adapter to the positive terminal of the battery, and the negative wire from the trailer adapter to the negative terminal of the battery. This will allow the battery to charge while the trailer is connected to a power source.

It's important to choose a battery that is appropriate for your needs and is designed for deep cycle use. You'll also need to make sure that the wiring between the battery and the lights is appropriately sized to handle the voltage and amperage of your system.

Overall, while it's possible to connect the battery directly to the trailer adapter, using a charge controller can help ensure safe and efficient charging and protect your battery and other electrical components.