Humanizing the conflict may help to neutralize some of the bitterness of the disagreement. One way to to this is to give a personal testimony, the story of how you arrived where you currently are. Also, everyone should be willing to wear their human fallibility on their sleeve. I think this helps to communicate open mindedness and a willingness to hear other people out.
What people hate more than anything is being talked down to which drives some of the anti-intellectualism that exists in these types of conflicts. When their is a difference in knowledge it is very difficult to avoid patronizing your audience which is why it may be helpful to discuss the basics of your reasoning. I have found that for science this can be a lot easier than people think since most of the public is actually very curious about science.