Disagreement is inevitable - perhaps how you react depends on what outcome you desire.

You may be able to change their mind in which case the 'how dare you approach' will diminish the chances of this, it is quite possible they are not susceptible to argument, so if you cannot change their views why waste energy on the 'How dare you's'?

It occurs to me that the 'How dare you response' is about you, not the other person or their views. Do you really need to validate yourself by exclaiming 'How dare you'?

In Britain we avoid all this by saying, 'Well, it's a point of view I suppose', thus expressing our disagreement in a non-judgemental or confrontational way and asserting the independence of our opinions.