No Christian should EVER play God's defense attorney in the blasphemous courtroom of the unsaved. When an unbeliever demands evidence for the existence of God or for the truth of His Word, evidence is not what they should be given.
While speaking the truth in love (Ephesians 4:15), a stern and uncompromising tone should be used as the Christian points out to the unbeliever that God's Word declares they do believe in God. They simply suppress the truth by their unrighteousness, for they love their sin and they love themselves more than they love the God they KNOW exists. This assertion should then be followed be declaring God's truth on the matter, Romans 1:18-19, as the authority by which the Christian has made the declaration.
All those--regardless of the depth of their scholastic abilities and debating prowess, regardless of their popularity in the evangelical, apologetic world--all those (pastor, street preacher, theologian, or layman) who believe it is a positive skill to prove the existence of God without referencing the truth of God's Word, in reality, show a lack of faith in the truth of God's Word. Likewise, those who believe it is necessary to prove the existence of God and the truth of His Word play the devil's advocate, not the Lord's advocate.
There is no evidence that demands a verdict. There is no case for Christ that must be made. God alone is the Judge, the Prosecuting Attorney, and the Jury. NO defense can ever be made against Him, and NO defense for unbelief will ever be sustained in His presence. God, His Word, and His Gospel are always positive assertions made by Christians, and they are assertions that must be made without compromise, without acquiescence, without apology, and without the slightest belief that such eternal truths could ever be lost in debate with sinful (Romans 3:23), God-hating (Romans 1:30), self-idolaters (2 Timothy 3:2).
No unbeliever ever has or ever will seek to "discover the God [their] heart is searching for," as one popular Christian apologist puts it; for the unregenerate heart never searches for God (Romans 3:11). The unregenerate human heart is deceitful above all else and desperately sick (Jeremiah 17:9). To help the unbeliever on some mythical, unbiblcal quest by providing them with evidences for belief in God is to merely assist them (Romans 1:32) on their hell-bent efforts to flee from the God they know exists (Proverbs 28:1).
No, Christian. Stop playing God's defense attorney in the blasphemous courtroom of the unredeemed. Instead, love the lost and love Jesus more than you love yourself and declare to the lost the truths they already know (Romans 1:19-23). Speak to them of the great mystery--a gospel for both Jew and Gentile alike (Ephesians 3:6; Ephesians 6:19). Trust in the power of the gospel to change and save lives (Romans 1:16)--a gospel that must be declared to be heard (Romans 10:14-17). Trust not in your own perceived ability to convince anyone to follow Christ. You have no such ability.
Don't let your well-intentioned efforts to prove God become a blasphemy against God.