In your example, "that" and "whether" serve different purposes.
"For many hours we discussed that we had to lower the price; however, we failed to reach a conclusion."
In this sentence, "that" is used to indicate a specific proposition or statement that was discussed. In this case, the discussion was focused on the proposition that they had to lower the price.
"For many hours we discussed whether we had to lower the price; however, we failed to reach a conclusion."
In this sentence, "whether" is used to indicate a question or doubt. In this case, the discussion was focused on whether or not they had to lower the price.