Answer: There is nothing in Scripture that teaches us that suicide is the unforgivable sin. I believe it has been the teaching of the Catholic church that all Suicides go to hell, but this isn't explicitly taught in the Bible.
If one wanted to build an assumption about suicide and salvation, it's not hard to imagine that suicide is not a good sign that a person is heaven-bound – since it is such a final and unrepentant act of rebellion against God. Also, no suicide in the Bible is of a righteous person (Saul, Judas, etc). And if it’s true that the New Birth leads to new life, then suicide is a horrible report card on whether the Birth has happened. It’s a not a good report card on a person's inner journey.
However, while all of that is true, and while the Bible teaches that a person who is heaven bound is born anew and as such should be increasing in their love for God and man (including themselves)... still, no person who experiences new life lives perfectly without sin. (1 John 1:8).
And thus you are correct in assuming that NO SIN is unforgivable... (with one exception, which I’ll comment on in a second) and that NO Christian is without sin. In one sense, a Christian dying while committing suicide is not terribly different than a Christian dying while committing adultery. Neither is a good way to go out but both have happened to true Christians before. If we believe that we are saved by grace through Christ and not by acts of righteousness (Rom 3:21.22), then it is faith and God's wonderful mercy and not any one act of sin that determines heaven or hell.
The idea that all suicides MUST go to hell is partly built on the faulty idea that a Christian is forgiven, and heaven-bound, but only if they can repent of each and every sin they commit after the New Birth. And obviously Suicide is an act that no one can repent of because it is their last act. But this idea of "YO-YO" salvation based on our moment by moment "performance" violates the very idea of Adoption which the Bible repeatedly uses as a descriptor for what happens in Salvation (Rom 8:23). If I am adopted into God's family, by grace, then one sin does not kick me out (1 John 2:1). In fact, nothing can keep me from the "love of God in Christ Jesus"... not hardship or danger or sword, or a bad week where I sin repeatedly. (Romans 8:28-39)
Now, there IS one sin that is unforgivable. Jesus said:
Matt 12:31 And so I tell you, every sin and blasphemy will be forgiven men, but the blasphemy against the Spirit will not be forgiven..."
So there IS an unforgivable sin – the sin against the Holy Spirit. Could that be suicide? What could that mysterious sin be? I believe this isn't as complicated as some make it. What does the Holy Spirit come to do? He brings the saving benefits of forgiveness to human hearts. So this means, everything is forgivable except this one thing: the rejecting of forgiveness. That’s a sin for which there is no forgiveness.
So we can say that suicide could only be the unforgivable sin IF it is a person's deliberate sin against the Holy Spirit, an final rejection of his saving benefits and healing touch. Then suicide marks a person for hell. But in all reality, that person was probably headed there already. Suicide isn't the magic act that condemns a person, but it may be the final outward sign of a continual, inward rejection of grace.
While we cannot make pronouncements of anyone's eternal destiny based on any one act of sin, (or many acts of righteousness for that matter!) we can say this: Christians should be living out new Life in increasing love and victory, but only God knows the heart. Suicide is, at best, a mark of a defeated Christian who did not access God's power for New Life in a critical way. At worst, however, it may be the outward expression of the spiritual death that hangs over this person's soul, evidence of still being dead in sin, and therefore a dark harbinger of their eternity. (Eph 1:1-6)
It's up to God to know what suicide means for any individual person. If they claimed to be a follower of Christ, their salvation was not determined by enough good works, or the absence of all sin. Thank God! It's about grace. And we shouldn’t assume any such individuals are in hell. However, grace works. So suicide, while not directly corresponding to damnation, is surely an ending no Christian who claims to have God's power at work in them would want to choose.