Where in Acts 10 do you find this idea that Cornelius was unregenerate? There's nothing in the chapter which suggests such an idea. Oh, right... you have to believe that Cornelius was unregenerate, because you believe in Baptismal Regeneration, and Cornelius was not yet baptized.
Well, that just goes to show how Romish Error can interfere with your personal understanding of perspicuous Scripture. There is no more evidence in Acts 10 that Cornelius was unregenerate than there is evidence that Noah or Abraham or Moses or Jonah's Repentant Ninevities were unregenerate, or for that matter the Believing Jews and Gentiles who died during Christ's earthly ministry but prior to His crucifixion and resurrection.
Now, it is true that, the Promised Messiah having come, it was fitting that those who had Faith in the Promises of God should be informed of the fulfillment of God's Promise; and God used Cornelius as a demonstration of the fact that this Good News should proceed also to the Gentiles; but the notion that Cornelius, whom Scripture calls a "devout" and "God-fearing" man in Acts 10:2, was Hell-bound prior to receiving the information that the God in whom he already trusted had indeed fulfilled His messianic promises... frankly, that's just silly. But that's where Romish error will take you.
One cannot be both regenerated and unregenerated at the same time.
Well, you've got that much right. That's an important point for you to comprehend. And that is something upon which I can build.
Now when you realize that the Old Testament Saints were regenerated prior to Baptism, you'll likewise realize that Cornelius was regenerated prior to Baptism, and the next thing you know, you'll realize that Baptismal Regeneration is a Biblically-unsound notion, and pretty soon... why, you're making progress, Hermann!! We'll make a Bible-believing Protestant of you yet.
No sensible Christian has any reason to hold a strange new doctrine not ever heard of until 1500 years after Christ rose from the dead.
Precisely correct, which is why we reject the AD 1564 Council of Trent. And given that you correctly say, "One cannot be both regenerated and unregenerated at the same time", it is my fond hope that, by God's grace, you will too.
Quite simply, he did not have the Holy Ghost (Acts 10.44), hence he did not have supernatural grace or supernatural faith, hence he was unregenerate and still in his sins. Peter promises Cornelius quite plainly that those "who believe in [Jesus] may receive forgiveness of sins." (Acts 10.43). But Cornelius was not yet forgiven, otherwise Peter's visit would have been superfluous. And the brethren later glorify God saying "Therefore to the gentiles also God has given repentance unto life." (Acts 11.18). Cornelius was quite plainly unregenerate, yet God calls him "devout and God-fearing" and notes the cause of his special blessing was his "prayer" and his "alms".
but the notion that Cornelius, whom Scripture calls a "devout" and "God-fearing" man in Acts 10:2, was Hell-bound prior to receiving the information that the God in whom he already trusted had indeed fulfilled His messianic promises
Okay, so your definition of regeneration does not involve reception of the Holy Spirit and forgiveness of sins, but can occur prior to that time? Now read St. Augustine's comments again:
"As, then, we ought not to depreciate a man's righteousness, which begins to exist before he is joined to the Church, as the righteousness of Cornelius began to exist before he was in the body of Christian men,--which righteousness was not thought worthless, or the angel would not have said to him, 'Thy prayers and thine alms are come up as a memorial before God;' nor did it yet suffice for his obtaining the kingdom of heaven, or he would not have been told to send to Peter ..." (St. Augustine, On Baptism, 4.21.29)
Doesn't sound very regenerate if he had not yet obtained the kingdom. This is no different from what St. John Chrysostom teaches:
See how great the virtue of alms, both in the former discourse, and here! There, it delivered from death temporal; here, from death eternal; and opened the gates of heaven. Such are the pains taken for the bringing of Cornelius to the faith, that both an angel is sent, and the Spirit works, and the chief of the Apostles is fetched to him, and such a vision is shown, and, in short, nothing is left undone. How many centurions were there not besides, and tribunes, and kings, and none of them obtained what this man did! (St. John Chrysostom, Homilies in Acts, 22 [AD 388])
If he was regenerate, why does St. John Chrysostom say he was "delivered from death eternal" by way of his alms causing St. Peter's visit? Why send for St. Peter to teach him the faith? Wouldn't he know it already? Isn't that in fact what you say here:
Now, it is true that, the Promised Messiah having come, it was fitting that those who had Faith in the Promises of God should be informed of the fulfillment of God's Promise; and God used Cornelius as a demonstration of the fact that this Good News should proceed also to the Gentiles
Sorry OP, but he did not have faith. You are just reading things into the text which are plainly not there in order to prop up your heresy. It says he was "devout and God-fearing" (Acts 10.2), but he did not have faith in Jesus Christ, which is why St. Peter preaches him the gospel (Acts. 10:34-43) Nor was the fulfillment of the prophets in Jesus unknown. Remember what St. Paul said to Festus and Agrippa? "... none of these things escaped him; for none of them happened in a corner." (Acts 26.26) St. Peter said it also to Cornelius. "You know what took place throughout Judea ..." (Acts 10.37). He knew it, but he did not yet believe, because to believe is a supernatural gift that can only come by hearing the Truth.
Now when you realize that the Old Testament Saints were regenerated prior to Baptism
At least for Jewish males, I believe it occurred at circumcision, and for females at their presentation in the Temple.