Is King Saul in perdition? If so, why?
Your question was the first I’d ever heard of Saul being in perdition. I did a little research into it, and could not find any Latter-day Saint opinion on it. The only voices speaking out on this matter (either for or against it) are those who lack the light of modern revelation. As such, they do not have a proper understanding of heaven and hell, and the spirit world to properly make such statements.
For the modern saint, “perdition” is a loaded term. It conjures up the image of a kingdom without glory where Satan and his disciples are destined to dwell. In order to qualify for it, individuals must “know [God’s] power, and have been made partakers thereof, and suffered themselves through the power of the devil to be overcome, and deny the truth and defy my power” (D&C 76:31). They must “den[y] the Holy Spirit after having received it, and … the Only Begotten Son of the Father, having crucified him unto themselves and put him to an open shame” (D&C 76:35). Joseph Smith summarized it thusly:
“All sins shall be forgiven, except the sin against the Holy Ghost; for Jesus will save all except the sons of perdition. What must a man do to commit the unpardonable sin? He must receive the Holy Ghost, have the heavens opened unto him, and know God, and then sin against him. After a man has sinned against the Holy Ghost, there is no repentance for him. He has got to say that the sun does not shine while he sees it; he has got to deny Jesus Christ when the heavens have been opened unto him, and to deny the plan of salvation with his eyes open to the truth of it” (“Teachings of the Prophet Joseph Smith”, 358).
The closest we find King Saul having the heavens opened unto him or partaking of God’s power is when he joins the prophets for a brief stint (1 Samuel 10:9-13, and 19:20-24). This “new heart” doesn’t last long, as Saul usurps priestly authority, and even attempts murder. What’s apparent is that although Saul’s heart is filled with treachery, he is not past the influences of the Spirit. Thus, I wouldn’t classify him as a son of perdition.
The term “perdition”, outside of The Church of Jesus Christ of Latter-day Saints, is sometimes used as a substitute for “hell”. I suspect that’s where you’re getting this idea, so let’s shed some modern revelation on hell. In the final, ultimate sense, outer darkness is the hell where the devil and his minions go. In another sense, spirit prison is hell. It is the endless hell (where “endless” has a special meaning) where the unrepentant suffer for their sins until they accept Christ. As Saul died seeking revelation via witchcraft and necromancy (1 Sam. 28), it’s not unreasonable to assume that he passed from this world to the spirit world unrepentant and spent some time in this hell.