I'm curious why would you suggest there is such a thing as a Christian Atheist? Every atheist I know, have ever met and talked to online lacks a belief in a generic deity figure, never specifically singling out any specific deity. No atheist claims disbelief in deities of specific creeds; ALL GODS because of the fact that ANY creed represented by a deity are equally scrutinized on their existence.
So if Islam is truly different, and I admit I know very little about it (and I'm not asking right now to point to a specific example) but do you strongly believe there is something for an atheist to learn that would satisfy their requirement of concrete evidence that everything about it is real and true? If you do believe that, then I happily accept the challenge and I will learn about it. But right off the bat, what I got from the OP post is essentially a threat that once I have taken in an interest in Islam to reject it would have dire consequences. Do you not understand how that probably more than anything is extremely unconvincing and a guaranteed turnoff to any atheist? Threats in any argument almost always invalidate any shed of possibility that a logical, well-reasoned discussion can even take place.
Well, I suggested Christian Atheist because you rejected religion based off of the faith of Christianity, not any other religion. You are an Atheist like any other nonetheless. My pointing out you were Christian was to make a point that you had a perspective of religion based on that view. There are Hindu Atheists, Muslim Atheists, etc. And those are only those who rejected those religions. Some are born Atheist and are only Atheist.
And as to knowing if Islam is concrete and true based on some evidence, i don't know if an Atheist who learned about Islam would consider it true or not. What I believe may be concrete evidence, and you might still reject it altogether based on your own understanding of things. To explain, imagine the concept of revelation from God. I have a friend who is Agnostic with whom I speak about religion more than almost anyone else I have ever known (quite literally). Yet, despite anything I tell him, the one thing he does not give the time of day and considers to be so outlandish as to not even be worth his time, is the idea that God speaks to man. He things all of that is pure hocus pocus and will never listen to anything like that. Yet, religion is based purely off of Prophethood, the very fact that God spoke to the Prophets of old.
Furthermore, what these Prophets spoke of was the unseen, things which cannot be known to man, yet they are given as knowledge to them by God. This could mean something which happens and they were not present at but they learned of it, a future event, a fact of life or a teaching and story from a past religion they never knew of, etc. Other things include miracles and certain supernatural instances which no magician or person can fake but which are considered true for whatever reason. If you give the idea of Prophethood a chance, then you can engage in a meaningful discussion. But if you are one of those people who would never want to talk about it, then you will not be able to understand Islam at all. The whole religion was revealed by God. If you take that away, then you will never understand it and then you cannot consider yourself open-minded in a discussion. I don't consider my Agnostic friend open-minded, though he likes to assume he is.
As for threats of being punished for rejecting Islam, I don't know why you would worry about it since you don't believe in a God in the first place. That threat shouldn't mean anything to you. I had another friend who I told about the punishment of Hell (based on a question she asked), and she said she wouldn't mind Hell. I told her it is a terrible place, but she said she would still enjoy it. I told her good luck and enjoy being burnt. She still didn't get the idea that Hell is made to be a place we don't like. So, my point is that if you don't believe in it, then it shouldn't matter. But I agree that it is a big turnoff for discussion.
As for a well-reasoned logical discussion, we would have to point out from the very beginning what entails a well-reasoned and logical discussion. I already told you that it has to be open-minded. We would have to know what the purpose in mind is as well. If it is for you to understand Islam better, then you have to open up to new perspectives. If it is on the idea of Islam and how an Atheist perceives it, then it would be philosophical and we on this forum would have to adapt (although, since we are Muslim we have certain limits on what we can say so that we do not displease God). That being said, you explain the purpose of a discussion you want, and then we can go over what would constitute a well-reasoned logical discussion.
If anything I said wasn't clear, let me know. I am just about to go to sleep so I might be a little off in my thinking right now.