Some great questions to get started:

• Read 2 Timothy 3:16-17, Proverbs 30:5, and Isaiah 40:8. Where is your standard of truth… Is it the Bible?
• Read 1 John 4, 1 Thessalonians 5:21, Acts 17:10-11, 2 Corinthians 11:13-15 and Revelation 2:2: Shouldn’t we put Branham under the test? If not, why?
• Hypothetically, If the Bible said a doctrine differently than Branham, who is right?
• If the Bible clearly disproved Branham was who he says he was, would you consider stopping following him?
• Read Jeremiah 17:9, 2 Timothy 4:3, Proverbs 12:15, and Proverbs 28:26 If something feels right, does that make it right? 
• Read Proverbs 6:16-19, Proverbs 19:9, and Colossians 3:9-10. If Branham contradicts himself, or tells stories drastically differently, does that mean some of the things he says are false or possibly lies?
• Read Matthew 7:15-20, Galatians 5:22-23, Colossians 3:12, and 1 Corinthians 13:1-5. Do the leaders of your church show fruit of the spirit?

That starting point will help you see if they can even have a discussion about it. If they respond with some kind of dismissal then I don’t know if now is the right time to have a discussion.

If they answer the above questions reasonably, you can dig a little deeper, one point at a time. Again, don’t overload someone with facts. Move one at a time and get them thinking, you can always pause and continue at another time.