Below are three novel ideas for helping you gain rapport. Rapport isn't all or nothing. There are many degrees of rapport. Each of these can help you gain a greater degree. Practice them one at a time:
1. Notice the person's body posture and make your body posture more like his. Read more about this.
2. Try to use the same phrases he uses. For example, if he calls heterodox Muslims "moderate Muslims" and you happen to hate that term, put aside your personal prejudices for the moment and while talking to him, use his phrase, not yours, when referring to heterodox Muslims.
3. Match the speed of the person's speech. If she talks quickly, and you like to talk slow, you can gain more rapport if you speed up the pace of your speech to more closely match hers. Read more about this.
All these things can help the other person feel more connected to you, and it will actually make you feel more connected to her too. And this will make your ideas have a better chance of finding fertile soil to take root in. It will help open her mind to what you're saying. Attain and maintain rapport with her. It will go a long way to helping you get through.
For more ideas about how to talk to people about Islam, check this out: Tools to Help You Educate Your Fellow Non-Muslims About Islam.