The only Protestant denomination I have found that gives unofficial advice about sharing the gospel with atheists or agnostics is the Free Church of Scotland. Professor Donald Macleod, Systematic Theology Professor at the Free Church of Scotland College, takes an uncompromising stand and has this to say:
- No Christian can accept peoples' claims to be atheists or agnostics. Such claims contradict what the Bible tells us about human nature. Wherever we go, Christians have to assume that there is a divinely implanted awareness of God in every human soul.
- Christians can take it for granted that basic religious concepts such as God, eternity, holiness, sin and judgment are meaningful to all men and women. To begin our evangelism by trying to get behind this framework (by setting out to prove that God exists) is to walk into a philosophical maze. Christians should not confuse knowledge of God (religion) with knowledge of how we know Him (epistemology). Unfortunately, on here a lot of Christians make that mistake.
- All Christian witness must start from the assumption that 'the seed of religion' exists in every human heart. All people have a sense of the holy, of the spiritual. Everyone has a feeling of dependence. All have God's law within their hearts. They all know that they are answerable to God. No matter how loudly they protest to the contrary! Christians are not called to instil these instincts into people. Our responsibility is to build on the foundation which God has already laid.
Source: 'Behold Your God' by Donald Macleod pp 26-27 (Christian Focus 1990); Systematic Theology Professor at the Free Church of Scotland College
During my search for relevant information, I found two quotes from Aiden W. Tozer which speaks to your comment that “the only thing that can turn [atheists] around would be a supernatural, "road to Damascus" kind of experience. I see no other way.”
“The Spirit never bears witness to an argument about Christ, but He never fails to witness to a proclamation of Christ crucified, dead and buried, and now ascended to the right hand of the Majesty on high… Light is not enough. The inward operation of the Holy Spirit is necessary to saving faith. The gospel is light but only the Spirit can give sight.” (Born After Midnight)
What it means to be crucified [Romans 6:6 & Galatians 2:20]: Self is the opaque veil that hides the Face of God from us. It can be removed only in spiritual experience, never by mere instruction. As well try to instruct leprosy out of our system. There must be a work of God in destruction before we are free. We must invite the cross to do its deadly work within us. We must bring our self-sins to the cross of judgment. (The Pursuit of God)
To answer your question: As far as I know, there is no Christian denomination that gives official advice on how to win atheists for Christ. I only know that Christians are urged to "speak the truth in love" to atheists and to all others, and not to get into arguments about words. That detracts from pointing people to "the simplicity that is in Christ" (2 Corinthians 11:3). Christians should care about atheists to the extent that they are prepared to share the good news of Jesus Christ with them, so that they might come to know the love of God. Christians can, and should, simply share with them the reality of Christ (their own testimony, which might not be believed but cannot be refuted) and use that to explain the gospel of Christ to them. The Holy Spirit will do the rest.
That’s because coming to faith in Christ is not a head-trip – it’s a heart transplant.
Additional information and Resources for further investigation: There are many eminently qualified and intellectual Christians who have publicly debated with atheists. A now-dead former atheist and intellectual famously declared that he was the most reluctant of converts to Christianity (C.S. Lewis in “Mere Christianity”). The Reverend D. Robertson (Free Church of Scotland) has blog sites and books, such as ‘The Dawkins Letters – Challenging Atheist Myths’ (Christian Focus 2007). Professor John C. Lennox has had public debates with atheists Christopher Hitchens (now dead), Richard Dawkins, Laurence Kruss, Stephen Law and Peter Singer. Lennox has written a book 'Gunning for God: A Critique of the New Atheism' (Lion, 2011). John Stott - 'Why I Am a Christian,' (IVP, 2003) p. 87 (a riposte to Bertrand Russell's earlier work, 'Why I Am Not a Christian')
I have read the book ‘36 Arguments for the Existence of God’ by Rebecca Goldstein (First published in U.S.A. in 2010 by Pantheon Books, a division of Random House Inc.) The purpose of Goldstein’s book is to refute the existence of God. The following article presents the case for God’s existence: https://peterkreeft.com/topics-more/20_arguments-gods-existence.htm
This article invites atheists to focus on truth, regardless of beliefs and clears up some misconceptions (the source is a Protestant evangelical organisation, not a denomination: https://www.gotquestions.org/atheist-Christian.html
The Skeptic’s Annotated Bible http://www.skepticsannotatedbible.com/ is dealt with in this article: https://www.gotquestions.org/skeptics-annotated-Bible.html The purpose of the Got Questions article is to point out the fallacies behind the Skeptic’s Annotated Bible.