Sin is sin, whether it is deliberate, unintentional, or resisted but its temptation turns out to be irresistible, leading to rationalisation and even denial that it was ‘really’ a sin. However, God disciplines his stubborn children, the Psalmist saying such a one is blessed! (Ps. 94:12). Such discipline is not at all the same as ‘losing’ salvation. Christians are told:
“Despise not the chastening of the Lord, nor faint when thou art
rebuked of him; For whom the Lord loveth, he chasteneth, and scourgeth
every son whom he receiveth, If ye endure chastening, God dealing with
you as with sons” Hebrews 12:5-7
That alone should clear up the matter; the very idea that God would not correct his children, but allow some sin, hidden to his child, to only come out on the Day of Judgment, so as to damn that one! What kind of a god would do that?
You give two points as “teachings supported by Scripture”, with lists of Scripture references for both. A problem is with the latter half in point 2, namely, that “sinners who sin unknowingly are ‘poisoned by bitterness and bound by iniquity’ until they repent and pray for forgiveness (Acts 8:20-24”). In the text used as support, that person did not know he was poisoned by bitterness and bound by iniquity until the apostle Peter was given that insight (for Simon’s heart condition was revealed – God reads hearts, not men). Then, upon hearing the command to repent of his wickedness and to pray that the Lord might forgive him, Simon the sorcerer besought Simon Peter to pray for him.
It is possible for a believer to do something sinful while feeling very cheerful and convinced that they had not sinned. This can depend on how much they know about the high calling to holiness in word, deed and thought. Christians should become more sensitive to sin as they progress in their Christian lives. Many will say they learned with dismay how many things they used to do for years as new Christians that now they abhor and have learned to turn from. Yet there is an important point about not repenting and praying for forgiveness which can lead to the development of bitterness, and being bound by sin. This is crucial to the answer being sought, and although I can speak from a personal experience 17 years ago, I will not detail that on a public forum.
A Christian has the indwelling of the Holy Spirit, who convicts of sin. It was this work of the Holy Spirit that first led a person to repent of sin and to place all their faith in Christ for salvation. Once a Christian, they become more sensitive to sin as they desire to be more and more like Jesus. Ideally, they will make steady progress but sometimes it’s a case of one step forward, two back. That’s when the Holy Spirit disturbs, until there is prayerful seeking, repentance, and going forward in faith again. Then the peace of God returns.
But if a Christian turns a blind eye to their sin, dismissing it as slight, or rationalising it away with excuses, that is when bitterness can start to take root in their heart, as it did with me, despite several warnings before eventual discipline. Only after God’s discipline fell did I truly face up to my sin, confessing the truth of Hebrews 12:15.
When quoting 1 John 5:18, as to how Christians can know that they have eternal life (present tense), John links that assurance to having “believed in the name of the Son of God”. This is the key to assurance. If the expression of belief is the same as that of the apostle Peter in Matthew 16:13-17, then it will not be a verbal, formulaic response but an exclamation of faith given by the Father’s revelation to us. That’s how it was with Peter, and Jesus rejoiced that Peter’s new understanding had been given to him by God. None of the other apostles (up to that point) had twigged as to just what it meant to believe in Jesus as the Son of God. That expression of belief in the name of the Son of God gets to the core of understanding just who Jesus really is. That is why so many people who sincerely say they are Christians shrink back from expressing assurance about the promise of salvation: the full reality of just who Jesus is has not yet broken through, as it did with Peter. But when that happens, they will know the power of that transforming grace of God, in Christ, and all their confidence will be in that, and not in themselves, either trying (vainly) to live a sin-free life, or diligently and fearfully seeking to confess every sin, known or (impossibly) unknown to them.
Finally, all who trust only in what Christ did to save them should know even now that all their sin was freely pardoned due to the magnitude of what Christ achieved. But on the Day of Judgment all those who followed mere religious systems or formulas, or who were totally irreligious, will discover they are “the goats”, and all their sin will be punished. Their unsaved state will remain. The Righteous Judge of all the earth will judge justly. Everything hidden will be revealed.
So, in answer to, Can punishment for unintentional sins be reconciled with Christians knowing they have eternal life? - It is only those who are not yet children of God who should fear adverse judgment. As the Bible states,
“The Spirit itself beareth witness with our spirit, that we are
children of God: And if children, then heirs; heirs of God, and
joint-heirs with Christ; if so be that we suffer with him, that we may
be glorified together.” (Romans 8:16-17)
The reconciliation lies in Christians being disciplined by God, before the Day of Judgment, for he knows all their sin, and because they walk in the light of Christ, they see the sin they need to confess, exactly as 1 John 1:6-9 explains.