It does seem that you are suggesting that Paul would only ever say 'Christ died for our sins' if he has already established his audience to be only believers.
I have, by my count, three times now given you a hypothetical context wherein Paul could have said "Christ died for our sins" in a particularly redemptive sense to unbelievers. All it would take is for Paul to have used one exclusive pronoun, when he spoke on behalf of him and the church together as "we." I see no reason to accept that your hypothetical context as necessary while you conversely reject my hypothetical context out of hand.
Paul has mislead us then. As I have already said, Paul says in his letter, '...this is what we preach, and this is what you believed.' I don't know how to make it any clearer.
That line of argumentation is as clear as mud. I know you can't make it any clearer, but that's because your argument is indefensible, not because you've already made it sufficiently clear.
Say I give a lecture to a room full of Americans and Japanese about the Battle of Midway. Years later one of the Americans and I meet again, and he vaguely recalls this lecture, but doesn't remember who won. I want to refresh his memory, so I say, "This is what I taught, that we won." There is no grammatical or morphological or syntactical or metasyntactical justification for either taking that as a direct quote or for insisting that the antecedent of "we" includes anyone more than the first and second person - myself and the American student.
You pointed to Paul's reference to people saved by the Gospel as examples of unbelievers.
Obviously, for you, it is absolutely critical that no preacher should ever imply that unbelievers are included in such a statement as 'Christ died for our sins'. Paul did not share your concerns. Paul does not guard against that which you so relentlessly and vehemently guard against.
As I have stated, repeatedly, I don't have a problem if the Bible includes unbelievers under Christ's death, as that only signifies there is an additional, non-atoning universality within Christ's death. It does not make universal the redemption which is inherent in reconciliation with the father which is inherent in the payment for sins which is effected by the death of Christ.
But moreover, you need to go back and read and at least attempt to interact with what I've written on second person clusivity. You've demanded on several occasions that I explain why Paul isn't guarding against a certain misunderstanding, but I have already established that the human brain is unable to make that misunderstanding, because second person clusivity is too complicated a concept for the language center to process when handling a pronoun. In other words, the concept "not just me, not just you, but them also" needs to be present in a text for reasons beyond pronoun choice. It's not. It's fairly apparent that the reason you see it in this text is that you introduced it to this text because of your precommitment to general atonement.
Also consider, what I'm "vehemently defending against" is the idea that I Cor 15 has general atonement undertones, so it would be silly for me to expect Paul to explicitly preclude general atonement in this passage, since he's said nothing which would indicate it except to those who are already precommitted to it.