Sorry, I meant to do that. I found that the two reds were both positive. The black and white cables were both negative. You can see the final connection in the image attachment.
I first used the information from f5moab and bikendan to make the initial connection. I honestly thought that I remembered the two reds on opposite posts. However, after connecting the red (w/ breaker) and black cable to + and remaining white and red cables to -, everything stayed dead. I also noticed that the negative post got really hot to the touch and quickly disconnected everything.
I set out to do a closer inspection of the cables. I removed the bottom cover under the winch and found that the red (w/ breaker) and black cables were both connected to the winch motor. You can see this in the other image. That meant the black needed to be negative since the red one with the breaker had to be positive. Since the white was connected to the trailer frame, I was left with the "pale" red cable. Since the camper power still needed a connection, I deduced that the orphaned red cable must be positive.
After making the connection, everything worked perfectly. I made sure nothing was getting hot and kept an eye on it just in case I missed something. I haven't had any issues, so the final result was indeed the correct configuration.