So Cornelius is a person who seeks God but is without knowledge concerning Jesus and the Way.
This simply is not true. If it is true, then Peter is a liar because he said:
Acts 10:34-38
"Then Peter opened his mouth, and said, Of a truth I perceive that God is no respecter of persons: But in every nation he that feareth him, and worketh righteousness, is accepted with him.
The word which God sent unto the children of Israel, preaching peace by Jesus Christ: (he is Lord of all) That word, I say, ye know, which was published throughout all Judaea, and began from Galilee, after the baptism which John preached; How God anointed Jesus of Nazareth with the Holy Ghost and with power: who went about doing good, and healing all that were oppressed of the devil; for God was with him."
While Peter's order is true to the pattern (i.e. water baptism) used by John the Baptist, and continued by Jesus disciples for the Jewish nation (although not by Jesus see John 4:1-2). Is he being true to the direct command in the vision of Acts 10:11-16 not to call anything impure that God has made clean?
Why wouldn't he be true to it? God, in his foreknowledge, or because he foreknew how Cornelius and those assembled together with him would respond to Peter's gospel presentation, referred to these specific Gentiles to whom Peter was being sent as being "clean". God confirmed it by having the Holy Ghost fall upon them as they apparently believed what Peter was telling them, and then Peter water baptized them. Normally, water baptism would precede any type of Holy Ghost baptism, but you need to understand that Peter and other Jews at that time thought that salvation was only for the Jews. God corrected their theology in this special incident by putting his outward seal upon them first or before they were water baptized. Had they not had the Holy Ghost fall upon them first, then Peter might have forbidden them water baptism, and this is implied by Peter's statement:
Acts 10:47-48
"Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we? And he commanded them to be baptized in the name of the Lord. Then prayed they him to tarry certain days."
Notice the words "can any man forbid water?" In other words, the implication is that God was not only correcting Peter's theology, but also the theology of the other Jews who were assembled there with him.
Read Acts chapter 11. Peter initially got grilled for being with these Gentiles when he returned to Jerusalem, so God was also correcting the theology of those Jews as well.
Furthermore, even after this incident, Paul had to rebuke Peter to his face at Antioch because Peter was still incorrectly making a distinction between Jews and Gentiles, and that is after God had previously shown him the vision that you mentioned thrice.
Anyway, there is no reason whatsoever to believe that Cornelius and those assembled with him were unsaved when the Holy Ghost fell upon them. Apparently, they believed Peter's message before the Holy Ghost fell upon them, and God simply reversed the normal order of water baptism followed by Holy Ghost baptism because of the special circumstances surrounding this one incident.