Is the idea of soul sleep for would-be saints Biblical?
I write as a protestant, but I cannot speak for all protestants, but I will try to represent what I believe is a protestant majority view. (And my definition of "protestant" excludes those who do not believe in the Trinity: I cannot speak for them.)
First, though, you have packed a lot of ideas in your question which reflect a Catholic view which any protestant would feel a need to comment on and give a protestant view:
Protestants do not believe in purgatory
We do not believe in purgatory. We do not believe there is any perfecting to be completed after death by such means, and cannot believe it accords either with God's character, or with the effectiveness of Christ's finished work, or with scripture.
For the believer, there is a perfection made after death, but it is made in the instant of death by the grace of God. It is part of the salvation purchased by the finished work of Christ.
He has paid for all our sins so there is no payment to be made; he has imputed to us his righteousness so there is no unrighteousness to be cleansed; though we know we are still sinners in own fallen selves until the day we die, and though there are degrees of glory in the life to come depending on our obedience here, our sins are no barrier to us entering immediately into paradise upon death. Our sinful nature does not remain with us when we die. I myself serve the law of God (Romans 7:25); that part of me which serves sin I am freed from upon death.
We are no longer under any covenant of works so there cannot be any sins against it. When we believed we died to the moral law which brings condemnation (Romans 7:1-6). There is therefore no condemnation to those who are in Christ Jesus (Romans 8:1). When a believer dies he does not go to purgatory.
All believers in Christ are saints
The protestant view is also different about the word "saint". It means a believer in Christ, all of them are saints... First of all, there are not just a few, there are tens of thousands of them (Jude 1:14)! And how and when did the Apostle Paul become a saint? Answer: When "he was breathing out threats and slaughter" (Acts 9:1). It was then that the Lord met with him and turned his heart: from that time he was a saint, saved by the mighty grace of God.
There were saints everywhere in the New Testament church. Ending his letter to the Philippians Paul wrote "All the saints greet you, especially those that are of Caesar's household" (Phil 4:22). How many saints can you have in one household if the Catholic definition is assumed? But there were other saints in Rome also because "all the saints" were sending greetings whether they were of the household of Caesar or not. Moreover, they were sending greetings to the saints at Philippi: Paul asks that every saint at Philippi receives his greeting.. "Greet every saint in Christ Jesus" (Phil 4:21). By the Catholic definition this would mean that Paul chooses not to greet any believer unless they are a saint! Protestants conclude that all believers are saints by Paul's and the New Testament's definition. It is noteworthy that the word "saints" in the plural is used sixty times in the New Testament but only once in the singular and even here refers to many: "salute every saint" (Phil 4:21). The whole church at Rome was "called to be saints" (Romans 1:7). All believers are called to be set apart for the glory of God and our Lord Jesus Christ, to be living sacrifices for him (Romans 12:1,2; Luke 14:25-35). All true Christians are saints: there are no "would-be-saints" greeted in the New Testament, nor ever mentioned.
Do unbelievers go straight to hell?
Though it is not your question, briefly consider, that Peter said that Judas had "gone to his own place" (Acts 1:25): his soul had already gone there awaiting a bodily resurrection. Note also 1 Peter 3:19: the spirits which had been preached to (before the flood) are now "in prison". That is, their souls are now in hell awaiting the Day of Judgement when their bodies will be reunited with their souls and their sentence will be confirmed. And the rich man went to punishment before the Day of Judgement/Resurrection Day (Luke 16:22-23,26).
Do believers go straight to heaven?
Believers go straight to heaven when they die: this is the majority view of Protestants, though some have believed/believe in soul sleep, such as Bishop J.C. Ryle.
Our Lord Jesus said to a thief, and probably a murdering thief, "I tell you the truth, today you will be with me in paradise." (Luke 23:43).
In those last hours of his life he became a saint. He repented and believed in Jesus and was wonderfully saved. He didn't do anything except believe and repent and it came true for him just as Ezekiel said "if the wicked will turn from all his sins that he has committed and keep all my statutes and do that which is right he shall surely live, he shall not die. All his transgressions that he has committed, they shall not be mentioned to him" (Ezekiel 18:21-22). So upon death he went straight to glory.
Hebrews 12:18-24 assures us that right now "the general assembly and church of the firstborn" is "enrolled in heaven" (William Hendriksen, Bible on the Life Hereafter, page 51).
We are confident and willing rather to be absent from the body, and to be present with the Lord" (2 Corinthians 5:8). To die is to be separated from our body until the Day of Resurrection. In that state after death we shall be present with the Lord. What could be clearer?
The poor man went straight to Abraham's bosom (Luke 16:23) while others on earth were still alive.. ("Please, let me go and warn my brothers"). In the parable our Lord is using an idiom of the afterlife familiar with the Jews. If the idiom had been inaccurate Jesus would have not used it.
All the saints can and should "have a desire to depart and be with Christ, which is far better" (Philippians 1:23), though to abide is more needful for others.
There is no soul sleep. Believers go straight to heaven when they die. And I believe that this is the majority protestant view.
As the Westminster Confession of Faith says:
"The bodies of men, after death, return to dust, and see corruption: but their souls, which neither die nor sleep, having an immortal subsistence, immediately return to God who gave them: the souls of the righteous, being then made perfect of holiness, are received into the highest heavens, where they behold the face of God, in light and glory, waiting for the full redemption of their bodies. And the souls of the wicked are cast into hell, where they remain in torments and utter darkness, reserved to the judgment of the great day. Beside these two places, for souls separated from their bodies, the Scripture acknowledgeth none."