He misunderstands these verses, then…
1 Cor 15:51 Behold! I tell you a mystery. We shall not all sleep, but we shall all be changed.
Looking at 1 Cor 15:51 without looking at the following two verses is just… weird.
In v. 53, we see that in heaven, humans who had been corruptible and mortal will become incorruptible and immortal. In v. 52, we see that the dead (who were clearly mortal, and clearly corruptible) will be raised incorruptible. But, where does that leave the living, who are still mortal and still corruptible (but not yet dead)? Do they have to die first, so that they can be raised incorruptible (like those who had already died)? Paul is telling us that this isn’t the case: the dead will rise incorruptible, but they won’t be the only ones to put on immortality and incorruptibility; the living, too, will be changed “in the blink of an eye” and “at the last trumpet”. The dead proved their mortality long ago, and time has proven that they were corruptible; those who are alive at the end of time will not have to die and wait for their bodies to decay in order to attain to heaven.
So, no soul sleep when you look at it in context!
For David did not ascend to heaven, and yet he said, "'The Lord said to my Lord: "Sit at my right hand
Here, Peter isn’t trying to make any sort of claims about the state of a soul between death and the eschaton, let alone the claim that David is in ‘soul sleep’. Rather, Peter is trying to prove that Jesus is the Messiah. How does he do that? Well, first of all, Peter is relying on a few things that people of that day all agreed on: David was the author of the Psalms; David was God’s anointed King; David had been promised an eternal kingship through one of his descendants; and the things spoken of in Psalms refer to either David or to his descendant the Messiah.
So, Paul points to Psalm 110, in which we read “the LORD said to my Lord, ‘sit at my right hand’…”. Now, the first LORD refers to God – there’s no question about that. But, the question remains: who is ‘my Lord’? Is it David, or the Messiah? Well, there was nothing in Jewish tradition that asserted that David had ascended into heaven, so Peter asserts that this verse must be a prophecy of David about the Messiah! Therefore, the Messiah must be in God’s presence, seated at His right hand!
Peter has asserted that Jesus was not dead: they saw him resurrected. Therefore, if Jesus has been raised from the dead by God and if He is the Messiah, then he (not David) is seated at the right hand of God!
See what I mean? Peter is building a proof about Jesus being alive and in heaven, not about David being in ‘soul sleep’…! Now, he is right about David “not going to heaven right away” – after all, nobody was able to go to heaven until Jesus died on the cross and rose again! But, even Matthew (Mt 27:53) tells us that the saints entered the holy city after Jesus’ resurrection! So… David couldn’t get to heaven without Jesus (check). But there’s nowhere in the tradition (which is what Peter is appealing to) that suggests that David was in ‘soul sleep’…