From a Calvinist theological view, is it possible for God to predestine that XYZ be saved, but for XYZ to reject the doctrine of predestination throughout his or her life?
Without any question any person who never believes in predestination can still be considered a saved and regenerate predestined Christian by a Calvinist. All that is required to be considered 'predestined' in the eyes of men, is that a person believes in Christ for the forgiveness of their sins. Anyone who truly 'confesses Christ' in this sense will be saved, regardless of how clear, or confused they may be on the surrounding doctrines of their salvation.
If you declare with your mouth, “Jesus is Lord,” and believe in your heart that God raised him from the dead, you will be saved. 10 For it is with your heart that you believe and are justified, and it is with your mouth that you profess your faith and are saved. 11 As Scripture says, “Anyone who believes in him will never be put to shame.” 12 For there is no difference between Jew and Gentile —the same Lord is Lord of all and richly blesses all who call on him, 13 for, “Everyone who calls on the name of the Lord will be saved.” (NIV Romans 10:9-13)
Knowing how Calvinists leaders have thought through history, I can assure you if someone added a requirement like 'faith in the doctrine of predestination' to be needed in addition to 'faith in Christ' they would have denounced it. It is a 'person' who died for our sin that we must put our faith in, not a textbook statement phrased by church leaders.
This is a common point of confusion for people who haven't been exposed to much Reformed theology, so let me try to state this briefly and directly.
The Calvinistic doctrine of Predestination fits into a larger framework of doctrines, all built on scripture and playing off of each-other. If you try to transplant the doctrines like pieces from a puzzle and fit them into a different puzzle, they will stand out and look wrong.
In particular, our understanding of Predestination goes hand in hand with our understanding of salvation as a grace bestowed upon us by God. Just like we believe salvation cannot be earned through doing good works, we believe that it cannot be merited by believing the most sound doctrine. Quite simply this means that even people who actually hold some WRONG beliefs about God and salvation are going to be saved just because God wills it to be so.
On the flip side, we do believe that it pleases God for us to know how deeply we are indebted to his grace and that the closer we come to holding and promoting sound doctrine the more we will be transformed to be like him. While someone who insists on believing that it was their personal choice to follow God may well be saved by God's grace, an area of their life is held hostage to a kind of pride that thinks they accomplished something. One who deeply understands that they did not merit their salvation through their own actions or beliefs is setting the stage for a more humble and thankful life of grateful service.
In other words, belief in the doctrine of Predestination isn't necessary for salvation, but we do believe it serves a purpose in the life of a believer.
This perhaps isn't a typical answer to the question, but it does provide the opinion of one well-known Calvinist, George Whitefield.
Whitefield was once asked if he expected to see his Arminian friend John Wesley in heaven. He replied:
I fear not, for he will be so near the eternal throne and we at such a distance, we shall hardly get sight of him.
Cited in W. Wiersbe, Wycliffe Handbook of Preaching and Preachers, Moody Press, 1984, p. 255.