It is clear that you assume he modifies this gospel if an unbeliever is known to be present.
If Paul preached to a body of unbelievers an evangelistic message concerning Christ's definite redemption of a people taken out of all nations, when Paul later speaks to the portion of that body who are that people, it is entirely correct for him to say that Christ died for them. An unbelieving and unmentioned third party who might have been present during this message but to whom Paul is not speaking is not viable as an antecedent in this verse.
To sum up what you haven't proved in this thread thus far:
1)You haven't proven "Christ died for our sins" is a direct quote of what Paul spoke originally.
2)If "Christ died for our sins" was a direct quote, you haven't proven that Paul was speaking inclusively of the audience at the time. The context of the rest of the sermon would be necessary to determine this, and it wasn't recorded. Had Paul said "we" several times beforehand in sentences wherein he lumped himself together with the Corinthians, that would suggest inclusivity. Had he said anything along the lines of "I am an apostle sent from God and from the Church as an emissary unto you that you might believe and enter into the rest of the church with
us," that single use of exclusive "us" casts doubt on your entire thesis.
3)If the phrase is an indirect quote, you haven't proven that the antecedent extends beyond Paul and the Corinthian Church. Indeed, it cannot be the case that Paul intended to signify anything more than the inclusion of himself and the Corinthians. Pronouns which are explicitly inclusive of the second and third person do not exist. In any language. It is believed the language center in the brain isn't capable of handling that precise of a definition when trying to process a pronoun. To signify that you are speaking about a group containing both the second and third person requires explicit elaboration, and cannot be done with one pronoun.
4)Even if Christ died for the unsaved, you haven't proven that this constitutes the atonement. Since Christ did not efficaciously redeem the unsaved by ending their sin, all this means is that Christ's death contains an element besides the atonement which can be for all men, while the salvation inherent in Christ's bearing away of one's damnation remains merely for the saved. I am not endorsing this view, but it would be the logical necessity if your verse is taken the way you want it to be taken. Disputes over the atonement are first and foremost and always about what the atonement
is. If you are not prepared to argue from chapter and verse which explicitly state that the atonement is a potentiality, that's not exegetical theology. That's a contrivance designed to align the atonement with your perspectives on other doctrines.